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G = C4020(C2×C4)  order 320 = 26·5

10th semidirect product of C40 and C2×C4 acting via C2×C4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C85(C4×D5), C4020(C2×C4), C8⋊D55C4, (C4×D5).2Q8, C4.31(Q8×D5), C2.D810D5, C406C419C2, (C2×C8).65D10, C20.22(C2×Q8), C4⋊C4.171D10, C22.91(D4×D5), D10.17(C4⋊C4), C10.D820C2, C2.5(D8⋊D5), C20.Q820C2, C10.40(C8⋊C22), C53(M4(2)⋊C4), Dic5.18(C4⋊C4), (C2×C40).143C22, C20.109(C22×C4), (C2×C20).297C23, C2.5(Q16⋊D5), (C2×Dic5).223D4, (C22×D5).121D4, C10.69(C8.C22), C4⋊Dic5.123C22, C4.81(C2×C4×D5), C52C85(C2×C4), (D5×C4⋊C4).8C2, C2.16(D5×C4⋊C4), (C5×C2.D8)⋊7C2, C10.38(C2×C4⋊C4), (C4×D5).7(C2×C4), C4⋊C47D5.8C2, (C2×C8⋊D5).4C2, (C2×C4×D5).41C22, (C2×C10).302(C2×D4), (C5×C4⋊C4).90C22, (C2×C52C8).68C22, (C2×C4).400(C22×D5), SmallGroup(320,508)

Series: Derived Chief Lower central Upper central

C1C20 — C4020(C2×C4)
C1C5C10C2×C10C2×C20C2×C4×D5C2×C8⋊D5 — C4020(C2×C4)
C5C10C20 — C4020(C2×C4)
C1C22C2×C4C2.D8

Generators and relations for C4020(C2×C4)
 G = < a,b,c | a40=b2=c4=1, bab=a29, cac-1=a31, bc=cb >

Subgroups: 430 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, M4(2)⋊C4, C8⋊D5, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C10.D8, C20.Q8, C406C4, C5×C2.D8, D5×C4⋊C4, C4⋊C47D5, C2×C8⋊D5, C4020(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5, C22×D5, M4(2)⋊C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D8⋊D5, Q16⋊D5, C4020(C2×C4)

Smallest permutation representation of C4020(C2×C4)
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)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 61)(42 50)(43 79)(44 68)(45 57)(47 75)(48 64)(49 53)(51 71)(52 60)(54 78)(55 67)(58 74)(59 63)(62 70)(65 77)(69 73)(72 80)(81 89)(82 118)(83 107)(84 96)(86 114)(87 103)(88 92)(90 110)(91 99)(93 117)(94 106)(97 113)(98 102)(100 120)(101 109)(104 116)(108 112)(111 119)(122 150)(123 139)(124 128)(125 157)(126 146)(127 135)(129 153)(130 142)(132 160)(133 149)(134 138)(136 156)(137 145)(140 152)(143 159)(144 148)(147 155)(154 158)
(1 115 151 46)(2 106 152 77)(3 97 153 68)(4 88 154 59)(5 119 155 50)(6 110 156 41)(7 101 157 72)(8 92 158 63)(9 83 159 54)(10 114 160 45)(11 105 121 76)(12 96 122 67)(13 87 123 58)(14 118 124 49)(15 109 125 80)(16 100 126 71)(17 91 127 62)(18 82 128 53)(19 113 129 44)(20 104 130 75)(21 95 131 66)(22 86 132 57)(23 117 133 48)(24 108 134 79)(25 99 135 70)(26 90 136 61)(27 81 137 52)(28 112 138 43)(29 103 139 74)(30 94 140 65)(31 85 141 56)(32 116 142 47)(33 107 143 78)(34 98 144 69)(35 89 145 60)(36 120 146 51)(37 111 147 42)(38 102 148 73)(39 93 149 64)(40 84 150 55)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444444455888810···102020202020···2040···40
size11111010224444101020202020224420202···244448···84···4

50 irreducible representations

dim1111111112222222444444
type++++++++-++++++--+
imageC1C2C2C2C2C2C2C2C4Q8D4D4D5D10D10C4×D5C8⋊C22C8.C22Q8×D5D4×D5D8⋊D5Q16⋊D5
kernelC4020(C2×C4)C10.D8C20.Q8C406C4C5×C2.D8D5×C4⋊C4C4⋊C47D5C2×C8⋊D5C8⋊D5C4×D5C2×Dic5C22×D5C2.D8C4⋊C4C2×C8C8C10C10C4C22C2C2
# reps1111111182112428112244

Matrix representation of C4020(C2×C4) in GL6(𝔽41)

8100000
14330000
00136136
00536536
00405136
00365536
,
4000000
0400000
001000
00344000
000010
00003440
,
10200000
38310000
00250270
00025027
00270160
00027016

G:=sub<GL(6,GF(41))| [8,14,0,0,0,0,10,33,0,0,0,0,0,0,1,5,40,36,0,0,36,36,5,5,0,0,1,5,1,5,0,0,36,36,36,36],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[10,38,0,0,0,0,20,31,0,0,0,0,0,0,25,0,27,0,0,0,0,25,0,27,0,0,27,0,16,0,0,0,0,27,0,16] >;

C4020(C2×C4) in GAP, Magma, Sage, TeX

C_{40}\rtimes_{20}(C_2\times C_4)
% in TeX

G:=Group("C40:20(C2xC4)");
// GroupNames label

G:=SmallGroup(320,508);
// by ID

G=gap.SmallGroup(320,508);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,219,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^29,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations

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