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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 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×4], C22×C4 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8 [×2], C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, M4(2)⋊C4, C8⋊D5 [×4], C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C5×C4⋊C4 [×2], 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 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], 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 77)(42 66)(43 55)(45 73)(46 62)(47 51)(48 80)(49 69)(50 58)(52 76)(53 65)(56 72)(57 61)(59 79)(60 68)(63 75)(67 71)(70 78)(81 93)(83 111)(84 100)(85 89)(86 118)(87 107)(88 96)(90 114)(91 103)(94 110)(95 99)(97 117)(98 106)(101 113)(104 120)(105 109)(108 116)(115 119)(121 149)(122 138)(123 127)(124 156)(125 145)(126 134)(128 152)(129 141)(131 159)(132 148)(133 137)(135 155)(136 144)(139 151)(142 158)(143 147)(146 154)(153 157)
(1 44 82 160)(2 75 83 151)(3 66 84 142)(4 57 85 133)(5 48 86 124)(6 79 87 155)(7 70 88 146)(8 61 89 137)(9 52 90 128)(10 43 91 159)(11 74 92 150)(12 65 93 141)(13 56 94 132)(14 47 95 123)(15 78 96 154)(16 69 97 145)(17 60 98 136)(18 51 99 127)(19 42 100 158)(20 73 101 149)(21 64 102 140)(22 55 103 131)(23 46 104 122)(24 77 105 153)(25 68 106 144)(26 59 107 135)(27 50 108 126)(28 41 109 157)(29 72 110 148)(30 63 111 139)(31 54 112 130)(32 45 113 121)(33 76 114 152)(34 67 115 143)(35 58 116 134)(36 49 117 125)(37 80 118 156)(38 71 119 147)(39 62 120 138)(40 53 81 129)

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

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

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

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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