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G = C20⋊M4(2)  order 320 = 26·5

1st semidirect product of C20 and M4(2) acting via M4(2)/C2=C2×C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C201M4(2), C5⋊C82D4, C4⋊C4.6F5, C52(C86D4), C20⋊C84C2, C41(C4.F5), C2.12(D4×F5), C10.10(C4×D4), (C2×D20).12C4, D10⋊C810C2, C2.5(Q8.F5), D10⋊C4.3C4, C10.21(C8○D4), Dic5.71(C2×D4), D208C4.18C2, C10.14(C2×M4(2)), Dic5.56(C4○D4), C22.77(C22×F5), (C4×Dic5).192C22, (C2×Dic5).332C23, (C4×C5⋊C8)⋊4C2, (C5×C4⋊C4).9C4, C2.9(C2×C4.F5), (C2×C4.F5)⋊12C2, (C2×C4).26(C2×F5), (C2×C20).83(C2×C4), (C2×C5⋊C8).29C22, (C2×C4×D5).277C22, (C2×C10).43(C22×C4), (C2×Dic5).58(C2×C4), (C22×D5).49(C2×C4), SmallGroup(320,1043)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20⋊M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C4×C5⋊C8 — C20⋊M4(2)
C5C2×C10 — C20⋊M4(2)
C1C22C4⋊C4

Generators and relations for C20⋊M4(2)
 G = < a,b,c | a20=b8=c2=1, bab-1=a7, cac=a-1, cbc=b5 >

Subgroups: 474 in 122 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C86D4, C4×Dic5, D10⋊C4, C5×C4⋊C4, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×D20, C4×C5⋊C8, C20⋊C8, D10⋊C8, D208C4, C2×C4.F5, C20⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5, C86D4, C4.F5, C22×F5, C2×C4.F5, D4×F5, Q8.F5, C20⋊M4(2)

Smallest permutation representation of C20⋊M4(2)
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 98 27 47 146 62 111 133)(2 81 36 54 147 65 120 140)(3 84 25 41 148 68 109 127)(4 87 34 48 149 71 118 134)(5 90 23 55 150 74 107 121)(6 93 32 42 151 77 116 128)(7 96 21 49 152 80 105 135)(8 99 30 56 153 63 114 122)(9 82 39 43 154 66 103 129)(10 85 28 50 155 69 112 136)(11 88 37 57 156 72 101 123)(12 91 26 44 157 75 110 130)(13 94 35 51 158 78 119 137)(14 97 24 58 159 61 108 124)(15 100 33 45 160 64 117 131)(16 83 22 52 141 67 106 138)(17 86 31 59 142 70 115 125)(18 89 40 46 143 73 104 132)(19 92 29 53 144 76 113 139)(20 95 38 60 145 79 102 126)
(1 151)(2 150)(3 149)(4 148)(5 147)(6 146)(7 145)(8 144)(9 143)(10 142)(11 141)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(21 102)(22 101)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122222444444444458···8888810101020···20
size11112020224455551010410···10202020204448···8

38 irreducible representations

dim111111111222244488
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4C4○D4M4(2)C8○D4F5C2×F5C4.F5D4×F5Q8.F5
kernelC20⋊M4(2)C4×C5⋊C8C20⋊C8D10⋊C8D208C4C2×C4.F5D10⋊C4C5×C4⋊C4C2×D20C5⋊C8Dic5C20C10C4⋊C4C2×C4C4C2C2
# reps111212422224413411

Matrix representation of C20⋊M4(2) in GL6(𝔽41)

120000
40400000
000001
0040001
0004001
0000401
,
40390000
010000
0021822
0023162310
0031182518
0039393320
,
40390000
010000
000001
000010
000100
001000

G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,21,23,31,39,0,0,8,16,18,39,0,0,2,23,25,33,0,0,2,10,18,20],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;

C20⋊M4(2) in GAP, Magma, Sage, TeX

C_{20}\rtimes M_4(2)
% in TeX

G:=Group("C20:M4(2)");
// GroupNames label

G:=SmallGroup(320,1043);
// by ID

G=gap.SmallGroup(320,1043);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations

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