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G = C5×C22.50C24  order 320 = 26·5

Direct product of C5 and C22.50C24

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

Aliases: C5×C22.50C24, C10.1212- (1+4), C4⋊Q8⋊17C10, (Q8×C20)⋊36C2, (C4×Q8)⋊16C10, (D4×C20).28C2, (C4×D4).13C10, C22⋊Q8⋊18C10, C42⋊2C2⋊7C10, C42.50(C2×C10), C4.4D4.8C10, C42⋊C2⋊18C10, C20.347(C4○D4), (C2×C20).965C23, (C2×C10).376C24, (C4×C20).291C22, (D4×C10).325C22, C22.50(C23×C10), C23.21(C22×C10), (Q8×C10).277C22, C2.13(C5×2- (1+4)), (C22×C20).461C22, (C22×C10).104C23, (C5×C4⋊Q8)⋊38C2, C4.39(C5×C4○D4), C4⋊C4.76(C2×C10), C2.29(C10×C4○D4), (C5×C22⋊Q8)⋊45C2, (C2×D4).71(C2×C10), C10.248(C2×C4○D4), C22⋊C4.6(C2×C10), (C2×Q8).65(C2×C10), (C5×C42⋊2C2)⋊18C2, (C5×C42⋊C2)⋊39C2, (C5×C4⋊C4).401C22, (C22×C4).72(C2×C10), (C2×C4).38(C22×C10), (C5×C4.4D4).17C2, (C5×C22⋊C4).90C22, SmallGroup(320,1558)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.50C24
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C42⋊2C2 — C5×C22.50C24
Lower central
C1 — C22 — C5×C22.50C24
Upper central
C1 — C2×C10 — C5×C22.50C24

Subgroups: 282 in 212 conjugacy classes, 150 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×10], C2×C4 [×4], D4 [×2], Q8 [×6], C23 [×2], C10 [×3], C10 [×2], C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C20 [×4], C20 [×11], C2×C10, C2×C10 [×6], C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, C2×C20 [×3], C2×C20 [×10], C2×C20 [×4], C5×D4 [×2], C5×Q8 [×6], C22×C10 [×2], C22.50C24, C4×C20, C4×C20 [×6], C5×C22⋊C4 [×10], C5×C4⋊C4 [×2], C5×C4⋊C4 [×10], C22×C20 [×2], D4×C10, Q8×C10, Q8×C10 [×2], C5×C42⋊C2 [×2], D4×C20, Q8×C20, Q8×C20 [×2], C5×C22⋊Q8 [×2], C5×C4.4D4 [×2], C5×C42⋊2C2 [×4], C5×C4⋊Q8, C5×C22.50C24

Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×4], C24, C2×C10 [×35], C2×C4○D4 [×2], 2- (1+4), C22×C10 [×15], C22.50C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2- (1+4), C5×C22.50C24

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

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

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

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

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

Matrix representation ►G ⊆ GL4(𝔽41) generated by

1000
0100
00180
00018
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
40000
0100
0001
0010
,
0100
40000
00320
00032
,
9000
0900
0010
00040
,
32000
0900
0010
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,40,0,0,1,0,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,40],[32,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1] >;
 

125 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4S5A5B5C5D10A···10L10M···10T20A···20AV20AW···20BX
order1222224···44···4555510···1010···1020···2020···20
size1111442···24···411111···14···42···24···4

125 irreducible representations

dim11111111111111112244
type++++++++-
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10C4○D4C5×C4○D42- (1+4)C5×2- (1+4)
kernelC5×C22.50C24C5×C42⋊C2D4×C20Q8×C20C5×C22⋊Q8C5×C4.4D4C5×C42⋊2C2C5×C4⋊Q8C22.50C24C42⋊C2C4×D4C4×Q8C22⋊Q8C4.4D4C42⋊2C2C4⋊Q8C20C4C10C2
# reps12132241484128816483214

In GAP, Magma, Sage, TeX 

C_5\times C_2^2._{50}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.50C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1558);
 
// by ID
 
G=gap.SmallGroup(320,1558);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,436,1242,304]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=1,e^2=c*b=b*c,f^2=g^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,e*d*e^-1=b*d=d*b,g*e*g^-1=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,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀