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G = C5×C24.C22  order 320 = 26·5

Direct product of C5 and C24.C22

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

Aliases: C5×C24.C22, C2.7(D4×C20), C22⋊C4⋊5C20, (C2×C42)⋊2C10, (C2×C20).358D4, C10.138(C4×D4), C23.8(C2×C20), C24.2(C2×C10), C22.37(D4×C10), C2.C42⋊3C10, (C23×C10).2C22, C10.134(C4⋊D4), C10.64(C4.4D4), (C22×C20).33C22, C22.37(C22×C20), C23.61(C22×C10), C10.77(C42⋊C2), C10.33(C42⋊2C2), (C22×C10).452C23, C10.89(C22.D4), (C2×C4×C20)⋊3C2, (C2×C4⋊C4)⋊3C10, (C10×C4⋊C4)⋊30C2, C2.3(C5×C4⋊D4), (C5×C22⋊C4)⋊17C4, (C2×C4).34(C2×C20), (C2×C4).101(C5×D4), C2.2(C5×C4.4D4), (C2×C20).364(C2×C4), (C2×C10).604(C2×D4), (C2×C22⋊C4).6C10, C2.3(C5×C42⋊2C2), C22.22(C5×C4○D4), (C10×C22⋊C4).26C2, (C22×C10).87(C2×C4), (C22×C4).88(C2×C10), (C5×C2.C42)⋊5C2, C2.10(C5×C42⋊C2), (C2×C10).212(C4○D4), C2.5(C5×C22.D4), (C2×C10).325(C22×C4), SmallGroup(320,889)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C24.C22
Chief series
C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C10×C22⋊C4 — C5×C24.C22
Lower central
C1 — C22 — C5×C24.C22
Upper central
C1 — C22×C10 — C5×C24.C22

Generators and relations for C5×C24.C22
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=1, f2=e, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 322 in 190 conjugacy classes, 90 normal (62 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C24, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C24.C22, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C23×C10, C5×C2.C42, C2×C4×C20, C10×C22⋊C4, C10×C4⋊C4, C5×C24.C22
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22×C4, C2×D4, C4○D4, C20, C2×C10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×C20, C5×D4, C22×C10, C24.C22, C22×C20, D4×C10, C5×C4○D4, C5×C42⋊C2, D4×C20, C5×C4⋊D4, C5×C22.D4, C5×C4.4D4, C5×C42⋊2C2, C5×C24.C22

Smallest permutation representation of C5×C24.C22
►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)

(6 142)(7 143)(8 144)(9 145)(10 141)(16 154)(17 155)(18 151)(19 152)(20 153)(21 160)(22 156)(23 157)(24 158)(25 159)(31 128)(32 129)(33 130)(34 126)(35 127)(46 79)(47 80)(48 76)(49 77)(50 78)(61 75)(62 71)(63 72)(64 73)(65 74)(81 100)(82 96)(83 97)(84 98)(85 99)(86 123)(87 124)(88 125)(89 121)(90 122)(91 108)(92 109)(93 110)(94 106)(95 107)(101 131)(102 132)(103 133)(104 134)(105 135)(111 149)(112 150)(113 146)(114 147)(115 148)(116 139)(117 140)(118 136)(119 137)(120 138)

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

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

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

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

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

140 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4R5A5B5C5D10A···10AB10AC···10AJ20A···20AV20AW···20BT
order12···2224···44···4555510···1010···1020···2020···20
size11···1442···24···411111···14···42···24···4

140 irreducible representations

dim1111111111112222
type++++++
imageC1C2C2C2C2C4C5C10C10C10C10C20D4C4○D4C5×D4C5×C4○D4
kernelC5×C24.C22C5×C2.C42C2×C4×C20C10×C22⋊C4C10×C4⋊C4C5×C22⋊C4C24.C22C2.C42C2×C42C2×C22⋊C4C2×C4⋊C4C22⋊C4C2×C20C2×C10C2×C4C22
# reps12131848412432481632

Matrix representation of C5×C24.C22 ►in GL5(𝔽41)

10000
037000
003700
000160
000016
,
400000
01000
004000
00010
000140
,
10000
040000
004000
000400
000040
,
10000
040000
004000
00010
00001
,
400000
040000
004000
000400
000040
,
320000
003200
032000
000139
000140
,
10000
00100
040000
00090
00009

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,16,0,0,0,0,0,16],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,0,32,0,0,0,32,0,0,0,0,0,0,1,1,0,0,0,39,40],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,9] >;
 

C5×C24.C22 in GAP, Magma, Sage, TeX 

C_5\times C_2^4.C_2^2
 
% in TeX
 
G:=Group("C5xC2^4.C2^2");
 
// GroupNames label
 
G:=SmallGroup(320,889);
 
// by ID
 
G=gap.SmallGroup(320,889);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766,226]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=1,f^2=e,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^-1=b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
 
// generators/relations
 

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×
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