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G = C5×C22.57C24  order 320 = 26·5

Direct product of C5 and C22.57C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C22.57C24, C10.1722+ 1+4, C10.1242- 1+4, C4⋊Q8⋊18C10, C22⋊Q8⋊20C10, C42⋊2C2⋊9C10, C42.55(C2×C10), C42.C2⋊12C10, C4.4D4.9C10, (C2×C10).383C24, (C2×C20).684C23, (C4×C20).296C22, (D4×C10).225C22, C23.26(C22×C10), C22.57(C23×C10), (Q8×C10).188C22, C22.D4.3C10, C2.24(C5×2+ 1+4), C2.16(C5×2- 1+4), (C22×C20).463C22, (C22×C10).109C23, (C5×C4⋊Q8)⋊39C2, C4⋊C4.35(C2×C10), (C5×C22⋊Q8)⋊47C2, (C2×D4).38(C2×C10), C22⋊C4.8(C2×C10), (C2×Q8).31(C2×C10), (C5×C42.C2)⋊29C2, (C5×C42⋊2C2)⋊20C2, (C5×C4⋊C4).252C22, (C2×C4).43(C22×C10), (C22×C4).74(C2×C10), (C5×C4.4D4).18C2, (C5×C22⋊C4).93C22, (C5×C22.D4).6C2, SmallGroup(320,1565)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.57C24
Chief series
C1 — C2 — C22 — C2×C10 — C2×C20 — Q8×C10 — C5×C4⋊Q8 — C5×C22.57C24
Lower central
C1 — C22 — C5×C22.57C24
Upper central
C1 — C2×C10 — C5×C22.57C24

Generators and relations for C5×C22.57C24
 G = < a,b,c,d,e,f,g | a5=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Subgroups: 282 in 196 conjugacy classes, 142 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C20, C2×C10, C2×C10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22.57C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, Q8×C10, C5×C22⋊Q8, C5×C22.D4, C5×C4.4D4, C5×C42.C2, C5×C42⋊2C2, C5×C4⋊Q8, C5×C22.57C24
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, 2+ 1+4, 2- 1+4, C22×C10, C22.57C24, C23×C10, C5×2+ 1+4, C5×2- 1+4, C5×C22.57C24

Smallest permutation representation of C5×C22.57C24
►On 160 points
Generators in S160
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)

(1 35)(2 31)(3 32)(4 33)(5 34)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 28)(22 29)(23 30)(24 26)(25 27)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)(76 95)(77 91)(78 92)(79 93)(80 94)(81 88)(82 89)(83 90)(84 86)(85 87)(96 115)(97 111)(98 112)(99 113)(100 114)(101 108)(102 109)(103 110)(104 106)(105 107)(116 135)(117 131)(118 132)(119 133)(120 134)(121 128)(122 129)(123 130)(124 126)(125 127)(136 155)(137 151)(138 152)(139 153)(140 154)(141 148)(142 149)(143 150)(144 146)(145 147)

(1 30)(2 26)(3 27)(4 28)(5 29)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)(36 50)(37 46)(38 47)(39 48)(40 49)(41 53)(42 54)(43 55)(44 51)(45 52)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)(76 90)(77 86)(78 87)(79 88)(80 89)(81 93)(82 94)(83 95)(84 91)(85 92)(96 110)(97 106)(98 107)(99 108)(100 109)(101 113)(102 114)(103 115)(104 111)(105 112)(116 130)(117 126)(118 127)(119 128)(120 129)(121 133)(122 134)(123 135)(124 131)(125 132)(136 150)(137 146)(138 147)(139 148)(140 149)(141 153)(142 154)(143 155)(144 151)(145 152)

(1 115 35 96)(2 111 31 97)(3 112 32 98)(4 113 33 99)(5 114 34 100)(6 90 16 83)(7 86 17 84)(8 87 18 85)(9 88 19 81)(10 89 20 82)(11 95 156 76)(12 91 157 77)(13 92 158 78)(14 93 159 79)(15 94 160 80)(21 108 28 101)(22 109 29 102)(23 110 30 103)(24 106 26 104)(25 107 27 105)(36 135 55 116)(37 131 51 117)(38 132 52 118)(39 133 53 119)(40 134 54 120)(41 128 48 121)(42 129 49 122)(43 130 50 123)(44 126 46 124)(45 127 47 125)(56 155 75 136)(57 151 71 137)(58 152 72 138)(59 153 73 139)(60 154 74 140)(61 148 68 141)(62 149 69 142)(63 150 70 143)(64 146 66 144)(65 147 67 145)

(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 123 16 130)(7 124 17 126)(8 125 18 127)(9 121 19 128)(10 122 20 129)(11 116 156 135)(12 117 157 131)(13 118 158 132)(14 119 159 133)(15 120 160 134)(21 68 28 61)(22 69 29 62)(23 70 30 63)(24 66 26 64)(25 67 27 65)(36 95 55 76)(37 91 51 77)(38 92 52 78)(39 93 53 79)(40 94 54 80)(41 88 48 81)(42 89 49 82)(43 90 50 83)(44 86 46 84)(45 87 47 85)(96 136 115 155)(97 137 111 151)(98 138 112 152)(99 139 113 153)(100 140 114 154)(101 141 108 148)(102 142 109 149)(103 143 110 150)(104 144 106 146)(105 145 107 147)

(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 143 16 150)(7 144 17 146)(8 145 18 147)(9 141 19 148)(10 142 20 149)(11 136 156 155)(12 137 157 151)(13 138 158 152)(14 139 159 153)(15 140 160 154)(21 48 28 41)(22 49 29 42)(23 50 30 43)(24 46 26 44)(25 47 27 45)(56 90 75 83)(57 86 71 84)(58 87 72 85)(59 88 73 81)(60 89 74 82)(61 93 68 79)(62 94 69 80)(63 95 70 76)(64 91 66 77)(65 92 67 78)(96 123 115 130)(97 124 111 126)(98 125 112 127)(99 121 113 128)(100 122 114 129)(101 119 108 133)(102 120 109 134)(103 116 110 135)(104 117 106 131)(105 118 107 132)

(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)(76 95)(77 91)(78 92)(79 93)(80 94)(81 88)(82 89)(83 90)(84 86)(85 87)(96 103)(97 104)(98 105)(99 101)(100 102)(106 111)(107 112)(108 113)(109 114)(110 115)(116 123)(117 124)(118 125)(119 121)(120 122)(126 131)(127 132)(128 133)(129 134)(130 135)(136 150)(137 146)(138 147)(139 148)(140 149)(141 153)(142 154)(143 155)(144 151)(145 152)
 
G:=sub<Sym(160)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,115)(97,111)(98,112)(99,113)(100,114)(101,108)(102,109)(103,110)(104,106)(105,107)(116,135)(117,131)(118,132)(119,133)(120,134)(121,128)(122,129)(123,130)(124,126)(125,127)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147), (1,30)(2,26)(3,27)(4,28)(5,29)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72)(76,90)(77,86)(78,87)(79,88)(80,89)(81,93)(82,94)(83,95)(84,91)(85,92)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152), (1,115,35,96)(2,111,31,97)(3,112,32,98)(4,113,33,99)(5,114,34,100)(6,90,16,83)(7,86,17,84)(8,87,18,85)(9,88,19,81)(10,89,20,82)(11,95,156,76)(12,91,157,77)(13,92,158,78)(14,93,159,79)(15,94,160,80)(21,108,28,101)(22,109,29,102)(23,110,30,103)(24,106,26,104)(25,107,27,105)(36,135,55,116)(37,131,51,117)(38,132,52,118)(39,133,53,119)(40,134,54,120)(41,128,48,121)(42,129,49,122)(43,130,50,123)(44,126,46,124)(45,127,47,125)(56,155,75,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,148,68,141)(62,149,69,142)(63,150,70,143)(64,146,66,144)(65,147,67,145), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,123,16,130)(7,124,17,126)(8,125,18,127)(9,121,19,128)(10,122,20,129)(11,116,156,135)(12,117,157,131)(13,118,158,132)(14,119,159,133)(15,120,160,134)(21,68,28,61)(22,69,29,62)(23,70,30,63)(24,66,26,64)(25,67,27,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,88,48,81)(42,89,49,82)(43,90,50,83)(44,86,46,84)(45,87,47,85)(96,136,115,155)(97,137,111,151)(98,138,112,152)(99,139,113,153)(100,140,114,154)(101,141,108,148)(102,142,109,149)(103,143,110,150)(104,144,106,146)(105,145,107,147), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,90,75,83)(57,86,71,84)(58,87,72,85)(59,88,73,81)(60,89,74,82)(61,93,68,79)(62,94,69,80)(63,95,70,76)(64,91,66,77)(65,92,67,78)(96,123,115,130)(97,124,111,126)(98,125,112,127)(99,121,113,128)(100,122,114,129)(101,119,108,133)(102,120,109,134)(103,116,110,135)(104,117,106,131)(105,118,107,132), (6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,103)(97,104)(98,105)(99,101)(100,102)(106,111)(107,112)(108,113)(109,114)(110,115)(116,123)(117,124)(118,125)(119,121)(120,122)(126,131)(127,132)(128,133)(129,134)(130,135)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,115)(97,111)(98,112)(99,113)(100,114)(101,108)(102,109)(103,110)(104,106)(105,107)(116,135)(117,131)(118,132)(119,133)(120,134)(121,128)(122,129)(123,130)(124,126)(125,127)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147), (1,30)(2,26)(3,27)(4,28)(5,29)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72)(76,90)(77,86)(78,87)(79,88)(80,89)(81,93)(82,94)(83,95)(84,91)(85,92)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152), (1,115,35,96)(2,111,31,97)(3,112,32,98)(4,113,33,99)(5,114,34,100)(6,90,16,83)(7,86,17,84)(8,87,18,85)(9,88,19,81)(10,89,20,82)(11,95,156,76)(12,91,157,77)(13,92,158,78)(14,93,159,79)(15,94,160,80)(21,108,28,101)(22,109,29,102)(23,110,30,103)(24,106,26,104)(25,107,27,105)(36,135,55,116)(37,131,51,117)(38,132,52,118)(39,133,53,119)(40,134,54,120)(41,128,48,121)(42,129,49,122)(43,130,50,123)(44,126,46,124)(45,127,47,125)(56,155,75,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,148,68,141)(62,149,69,142)(63,150,70,143)(64,146,66,144)(65,147,67,145), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,123,16,130)(7,124,17,126)(8,125,18,127)(9,121,19,128)(10,122,20,129)(11,116,156,135)(12,117,157,131)(13,118,158,132)(14,119,159,133)(15,120,160,134)(21,68,28,61)(22,69,29,62)(23,70,30,63)(24,66,26,64)(25,67,27,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,88,48,81)(42,89,49,82)(43,90,50,83)(44,86,46,84)(45,87,47,85)(96,136,115,155)(97,137,111,151)(98,138,112,152)(99,139,113,153)(100,140,114,154)(101,141,108,148)(102,142,109,149)(103,143,110,150)(104,144,106,146)(105,145,107,147), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,90,75,83)(57,86,71,84)(58,87,72,85)(59,88,73,81)(60,89,74,82)(61,93,68,79)(62,94,69,80)(63,95,70,76)(64,91,66,77)(65,92,67,78)(96,123,115,130)(97,124,111,126)(98,125,112,127)(99,121,113,128)(100,122,114,129)(101,119,108,133)(102,120,109,134)(103,116,110,135)(104,117,106,131)(105,118,107,132), (6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,103)(97,104)(98,105)(99,101)(100,102)(106,111)(107,112)(108,113)(109,114)(110,115)(116,123)(117,124)(118,125)(119,121)(120,122)(126,131)(127,132)(128,133)(129,134)(130,135)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152) );
 
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,16),(7,17),(8,18),(9,19),(10,20),(11,156),(12,157),(13,158),(14,159),(15,160),(21,28),(22,29),(23,30),(24,26),(25,27),(36,55),(37,51),(38,52),(39,53),(40,54),(41,48),(42,49),(43,50),(44,46),(45,47),(56,75),(57,71),(58,72),(59,73),(60,74),(61,68),(62,69),(63,70),(64,66),(65,67),(76,95),(77,91),(78,92),(79,93),(80,94),(81,88),(82,89),(83,90),(84,86),(85,87),(96,115),(97,111),(98,112),(99,113),(100,114),(101,108),(102,109),(103,110),(104,106),(105,107),(116,135),(117,131),(118,132),(119,133),(120,134),(121,128),(122,129),(123,130),(124,126),(125,127),(136,155),(137,151),(138,152),(139,153),(140,154),(141,148),(142,149),(143,150),(144,146),(145,147)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(21,33),(22,34),(23,35),(24,31),(25,32),(36,50),(37,46),(38,47),(39,48),(40,49),(41,53),(42,54),(43,55),(44,51),(45,52),(56,70),(57,66),(58,67),(59,68),(60,69),(61,73),(62,74),(63,75),(64,71),(65,72),(76,90),(77,86),(78,87),(79,88),(80,89),(81,93),(82,94),(83,95),(84,91),(85,92),(96,110),(97,106),(98,107),(99,108),(100,109),(101,113),(102,114),(103,115),(104,111),(105,112),(116,130),(117,126),(118,127),(119,128),(120,129),(121,133),(122,134),(123,135),(124,131),(125,132),(136,150),(137,146),(138,147),(139,148),(140,149),(141,153),(142,154),(143,155),(144,151),(145,152)], [(1,115,35,96),(2,111,31,97),(3,112,32,98),(4,113,33,99),(5,114,34,100),(6,90,16,83),(7,86,17,84),(8,87,18,85),(9,88,19,81),(10,89,20,82),(11,95,156,76),(12,91,157,77),(13,92,158,78),(14,93,159,79),(15,94,160,80),(21,108,28,101),(22,109,29,102),(23,110,30,103),(24,106,26,104),(25,107,27,105),(36,135,55,116),(37,131,51,117),(38,132,52,118),(39,133,53,119),(40,134,54,120),(41,128,48,121),(42,129,49,122),(43,130,50,123),(44,126,46,124),(45,127,47,125),(56,155,75,136),(57,151,71,137),(58,152,72,138),(59,153,73,139),(60,154,74,140),(61,148,68,141),(62,149,69,142),(63,150,70,143),(64,146,66,144),(65,147,67,145)], [(1,75,35,56),(2,71,31,57),(3,72,32,58),(4,73,33,59),(5,74,34,60),(6,123,16,130),(7,124,17,126),(8,125,18,127),(9,121,19,128),(10,122,20,129),(11,116,156,135),(12,117,157,131),(13,118,158,132),(14,119,159,133),(15,120,160,134),(21,68,28,61),(22,69,29,62),(23,70,30,63),(24,66,26,64),(25,67,27,65),(36,95,55,76),(37,91,51,77),(38,92,52,78),(39,93,53,79),(40,94,54,80),(41,88,48,81),(42,89,49,82),(43,90,50,83),(44,86,46,84),(45,87,47,85),(96,136,115,155),(97,137,111,151),(98,138,112,152),(99,139,113,153),(100,140,114,154),(101,141,108,148),(102,142,109,149),(103,143,110,150),(104,144,106,146),(105,145,107,147)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,143,16,150),(7,144,17,146),(8,145,18,147),(9,141,19,148),(10,142,20,149),(11,136,156,155),(12,137,157,151),(13,138,158,152),(14,139,159,153),(15,140,160,154),(21,48,28,41),(22,49,29,42),(23,50,30,43),(24,46,26,44),(25,47,27,45),(56,90,75,83),(57,86,71,84),(58,87,72,85),(59,88,73,81),(60,89,74,82),(61,93,68,79),(62,94,69,80),(63,95,70,76),(64,91,66,77),(65,92,67,78),(96,123,115,130),(97,124,111,126),(98,125,112,127),(99,121,113,128),(100,122,114,129),(101,119,108,133),(102,120,109,134),(103,116,110,135),(104,117,106,131),(105,118,107,132)], [(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(56,75),(57,71),(58,72),(59,73),(60,74),(61,68),(62,69),(63,70),(64,66),(65,67),(76,95),(77,91),(78,92),(79,93),(80,94),(81,88),(82,89),(83,90),(84,86),(85,87),(96,103),(97,104),(98,105),(99,101),(100,102),(106,111),(107,112),(108,113),(109,114),(110,115),(116,123),(117,124),(118,125),(119,121),(120,122),(126,131),(127,132),(128,133),(129,134),(130,135),(136,150),(137,146),(138,147),(139,148),(140,149),(141,153),(142,154),(143,155),(144,151),(145,152)]])
 

95 conjugacy classes

class 1 2A2B2C2D2E4A···4M5A5B5C5D10A···10L10M···10T20A···20AZ
order1222224···4555510···1010···1020···20
size1111444···411111···14···44···4

95 irreducible representations

dim111111111111114444
type++++++++-
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C102+ 1+42- 1+4C5×2+ 1+4C5×2- 1+4
kernelC5×C22.57C24C5×C22⋊Q8C5×C22.D4C5×C4.4D4C5×C42.C2C5×C42⋊2C2C5×C4⋊Q8C22.57C24C22⋊Q8C22.D4C4.4D4C42.C2C42⋊2C2C4⋊Q8C10C10C2C2
# reps14212424168481681248

Matrix representation of C5×C22.57C24 ►in GL8(𝔽41)

10000000
01000000
00100000
00010000
000037000
000003700
000000370
000000037
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
103900000
004010000
004000000
014000000
000032000
00000900
00008090
000008032
,
3830000000
383000000
01512260000
381515290000
0000317390
00003410039
00002601034
0000026731
,
139000000
040000000
1400400000
1404000000
00000100
000040000
000003101
0000310400
,
10000000
01000000
104000000
100400000
00001000
00000100
0000317400
00003410040

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,39,40,40,40,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,32,0,8,0,0,0,0,0,0,9,0,8,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32],[38,38,0,38,0,0,0,0,30,3,15,15,0,0,0,0,0,0,12,15,0,0,0,0,0,0,26,29,0,0,0,0,0,0,0,0,31,34,26,0,0,0,0,0,7,10,0,26,0,0,0,0,39,0,10,7,0,0,0,0,0,39,34,31],[1,0,1,1,0,0,0,0,39,40,40,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,31,0,0,0,0,1,0,31,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,31,34,0,0,0,0,0,1,7,10,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
 

C5×C22.57C24 in GAP, Magma, Sage, TeX 

C_5\times C_2^2._{57}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.57C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1565);
 
// by ID
 
G=gap.SmallGroup(320,1565);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446,2571,436,6947,1242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=g^2=1,d^2=e^2=f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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