Copied to
clipboard

?

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, C10.1502+ (1+4), C10.1102- (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, C42.33(C2×C10), C10.82(C23×C4), C4.21(C22×C20), (C2×C10).340C24, (C2×C20).711C23, (C4×C20).276C22, C20.225(C22×C4), C22.3(C22×C20), C2.2(C5×2+ (1+4)), C2.2(C5×2- (1+4)), (D4×C10).333C22, C22.13(C23×C10), C23.32(C22×C10), (Q8×C10).285C22, (C22×C10).256C23, (C22×C20).443C22, (C2×C4)⋊5(C2×C20), (C10×C4⋊C4)⋊40C2, (C2×C4⋊C4)⋊13C10, (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, (C5×C4⋊C4).408C22, C22⋊C4.30(C2×C10), (C2×C4).57(C22×C10), (C22×C4).11(C2×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

Subgroups: 370 in 294 conjugacy classes, 242 normal (20 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C5, C2×C4, C2×C4 [×23], C2×C4 [×6], D4 [×12], Q8 [×4], C23 [×3], C10 [×3], C10 [×6], C42 [×6], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×8], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×C20, C2×C20 [×23], C2×C20 [×6], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C23.33C23, C4×C20 [×6], C5×C22⋊C4 [×6], C5×C4⋊C4, C5×C4⋊C4 [×9], C22×C20 [×9], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C10×C4⋊C4 [×3], C5×C42⋊C2 [×3], D4×C20 [×6], Q8×C20 [×2], C10×C4○D4, C5×C23.33C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C23×C4, 2+ (1+4), 2- (1+4), C2×C20 [×28], C22×C10 [×15], C23.33C23, C22×C20 [×14], C23×C10, C23×C20, C5×2+ (1+4), C5×2- (1+4), C5×C23.33C23

Generators and relations
 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 >

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 76)(2 77)(3 78)(4 79)(5 80)(6 110)(7 106)(8 107)(9 108)(10 109)(11 115)(12 111)(13 112)(14 113)(15 114)(16 103)(17 104)(18 105)(19 101)(20 102)(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 68)(42 69)(43 70)(44 66)(45 67)(46 64)(47 65)(48 61)(49 62)(50 63)(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)(121 148)(122 149)(123 150)(124 146)(125 147)(126 144)(127 145)(128 141)(129 142)(130 143)(131 137)(132 138)(133 139)(134 140)(135 136)

(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)

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

(1 35)(2 31)(3 32)(4 33)(5 34)(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)(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)

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

Matrix representation ►G ⊆ 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] >;
 

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+4)2- (1+4)C5×2+ (1+4)C5×2- (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

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
 

׿
×
⋊
ℤ
𝔽
○
≀