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G = C5×C23.33C23  order 320 = 26·5

Direct product of C5 and C23.33C23

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

Aliases: C5×C23.33C23, C10.1102- 1+4, C10.1502+ 1+4, C4○D4⋊5C20, (C4×D4)⋊6C10, D4⋊8(C2×C20), (C4×Q8)⋊6C10, Q8⋊7(C2×C20), (D4×C20)⋊35C2, (Q8×C20)⋊26C2, C2.9(C23×C20), C42⋊C2⋊7C10, C10.82(C23×C4), C4.21(C22×C20), C42.33(C2×C10), (C4×C20).276C22, (C2×C10).340C24, (C2×C20).711C23, C20.225(C22×C4), C22.3(C22×C20), C2.2(C5×2- 1+4), C2.2(C5×2+ 1+4), (D4×C10).333C22, C23.32(C22×C10), C22.13(C23×C10), (Q8×C10).285C22, (C22×C20).443C22, (C22×C10).256C23, (C2×C4)⋊5(C2×C20), (C2×C4⋊C4)⋊13C10, (C10×C4⋊C4)⋊40C2, (C2×C20)⋊39(C2×C4), (C5×C4○D4)⋊17C4, (C5×D4)⋊38(C2×C4), (C5×Q8)⋊34(C2×C4), C4⋊C4.83(C2×C10), (C2×C4○D4).9C10, (C10×C4○D4).23C2, (C2×D4).79(C2×C10), (C2×Q8).73(C2×C10), (C5×C42⋊C2)⋊28C2, C22⋊C4.30(C2×C10), (C5×C4⋊C4).408C22, (C22×C4).11(C2×C10), (C2×C4).57(C22×C10), (C2×C10).135(C22×C4), (C5×C22⋊C4).161C22, SmallGroup(320,1522)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C5×C23.33C23
Chief series
C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C22⋊C4 — D4×C20 — C5×C23.33C23
Lower central
C1 — C2 — C5×C23.33C23
Upper central
C1 — C2×C10 — C5×C23.33C23

Generators and relations for C5×C23.33C23
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=1, e2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf=bc=cb, bd=db, be=eb, bg=gb, cd=dc, geg-1=ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe >

Subgroups: 370 in 294 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C23.33C23, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C10×C4⋊C4, C5×C42⋊C2, D4×C20, Q8×C20, C10×C4○D4, C5×C23.33C23
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C20, C2×C10, C23×C4, 2+ 1+4, 2- 1+4, C2×C20, C22×C10, C23.33C23, C22×C20, C23×C10, C23×C20, C5×2+ 1+4, C5×2- 1+4, C5×C23.33C23

Smallest permutation representation of C5×C23.33C23
►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 76)(2 77)(3 78)(4 79)(5 80)(6 114)(7 115)(8 111)(9 112)(10 113)(11 102)(12 103)(13 104)(14 105)(15 101)(16 107)(17 108)(18 109)(19 110)(20 106)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 75)(37 71)(38 72)(39 73)(40 74)(41 67)(42 68)(43 69)(44 70)(45 66)(46 65)(47 61)(48 62)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 56)(96 156)(97 157)(98 158)(99 159)(100 160)(116 155)(117 151)(118 152)(119 153)(120 154)(121 147)(122 148)(123 149)(124 150)(125 146)(126 145)(127 141)(128 142)(129 143)(130 144)(131 137)(132 138)(133 139)(134 140)(135 136)

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

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

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

(1 35)(2 31)(3 32)(4 33)(5 34)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(96 115)(97 111)(98 112)(99 113)(100 114)(101 107)(102 108)(103 109)(104 110)(105 106)(116 135)(117 131)(118 132)(119 133)(120 134)(121 127)(122 128)(123 129)(124 130)(125 126)

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

170 conjugacy classes

class 1 2A2B2C2D···2I4A···4X5A5B5C5D10A···10L10M···10AJ20A···20CR
order12222···24···4555510···1010···1020···20
size11112···22···211111···12···22···2

170 irreducible representations

dim111111111111114444
type+++++++-
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C202+ 1+42- 1+4C5×2+ 1+4C5×2- 1+4
kernelC5×C23.33C23C10×C4⋊C4C5×C42⋊C2D4×C20Q8×C20C10×C4○D4C5×C4○D4C23.33C23C2×C4⋊C4C42⋊C2C4×D4C4×Q8C2×C4○D4C4○D4C10C10C2C2
# reps13362116412122484641144

Matrix representation of C5×C23.33C23 ►in GL6(𝔽41)

1800000
010000
001000
000100
000010
000001
,
4000000
0400000
000102
00400390
00040040
001010
,
100000
010000
0040000
0004000
0000400
0000040
,
100000
0400000
001000
000100
000010
000001
,
100000
090000
00182800
00282300
0023132313
0013181318
,
4000000
0400000
0040000
0004000
001010
000101
,
4000000
0400000
00400390
00040039
001010
000101

G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,1,0,0,1,0,40,0,0,0,0,39,0,1,0,0,2,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,9,0,0,0,0,0,0,18,28,23,13,0,0,28,23,13,18,0,0,0,0,23,13,0,0,0,0,13,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,1,0,0,0,0,40,0,1,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,1,0,0,0,0,40,0,1,0,0,39,0,1,0,0,0,0,39,0,1] >;
 

C5×C23.33C23 in GAP, Magma, Sage, TeX 

C_5\times C_2^3._{33}C_2^3
 
% in TeX
 
G:=Group("C5xC2^3.33C2^3");
 
// GroupNames label
 
G:=SmallGroup(320,1522);
 
// by ID
 
G=gap.SmallGroup(320,1522);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,891,2467,304]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=f^2=1,e^2=d,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e>;
 
// generators/relations
 

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