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G = C4.(C2×D20)  order 320 = 26·5

15th non-split extension by C4 of C2×D20 acting via C2×D20/C22×D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.64(C2×D20), (C2×C4).47D20, C4⋊C4.237D10, C20.144(C2×D4), (C2×C20).473D4, (C2×Dic10)⋊22C4, C10.Q1628C2, C42⋊C2.9D5, (C22×C10).77D4, C20.95(C22⋊C4), C20.125(C22×C4), (C2×C20).331C23, Dic10.42(C2×C4), (C22×C4).112D10, C53(C23.38D4), C23.55(C5⋊D4), C4.24(D10⋊C4), C2.2(D4.9D10), C10.106(C8.C22), (C22×C20).153C22, (C22×Dic10).13C2, C22.24(D10⋊C4), (C2×Dic10).271C22, C4.53(C2×C4×D5), (C2×C4).45(C4×D5), (C2×C20).266(C2×C4), (C2×C10).460(C2×D4), C10.87(C2×C22⋊C4), C22.74(C2×C5⋊D4), C2.19(C2×D10⋊C4), (C2×C4).242(C5⋊D4), (C5×C4⋊C4).268C22, (C2×C52C8).88C22, (C2×C4).431(C22×D5), (C2×C4.Dic5).19C2, (C2×C10).80(C22⋊C4), (C5×C42⋊C2).10C2, SmallGroup(320,631)

Series: Derived Chief Lower central Upper central

C1C20 — C4.(C2×D20)
C1C5C10C2×C10C2×C20C2×Dic10C22×Dic10 — C4.(C2×D20)
C5C10C20 — C4.(C2×D20)
C1C22C22×C4C42⋊C2

Generators and relations for C4.(C2×D20)
 G = < a,b,c,d | a4=c20=1, b2=a2, d2=cac-1=a-1, ab=ba, ad=da, bc=cb, dbd-1=a2b, dcd-1=ac-1 >

Subgroups: 494 in 150 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C23, C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×9], Dic5 [×4], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C52C8 [×2], Dic10 [×4], Dic10 [×6], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×C10, C23.38D4, C2×C52C8 [×2], C4.Dic5 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×6], C2×Dic10 [×3], C22×Dic5, C22×C20, C10.Q16 [×4], C2×C4.Dic5, C5×C42⋊C2, C22×Dic10, C4.(C2×D20)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8.C22 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.38D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4.9D10 [×2], C4.(C2×D20)

Smallest permutation representation of C4.(C2×D20)
On 160 points
Generators in S160
(1 140 23 106)(2 107 24 121)(3 122 25 108)(4 109 26 123)(5 124 27 110)(6 111 28 125)(7 126 29 112)(8 113 30 127)(9 128 31 114)(10 115 32 129)(11 130 33 116)(12 117 34 131)(13 132 35 118)(14 119 36 133)(15 134 37 120)(16 101 38 135)(17 136 39 102)(18 103 40 137)(19 138 21 104)(20 105 22 139)(41 89 142 68)(42 69 143 90)(43 91 144 70)(44 71 145 92)(45 93 146 72)(46 73 147 94)(47 95 148 74)(48 75 149 96)(49 97 150 76)(50 77 151 98)(51 99 152 78)(52 79 153 100)(53 81 154 80)(54 61 155 82)(55 83 156 62)(56 63 157 84)(57 85 158 64)(58 65 159 86)(59 87 160 66)(60 67 141 88)
(1 130 23 116)(2 131 24 117)(3 132 25 118)(4 133 26 119)(5 134 27 120)(6 135 28 101)(7 136 29 102)(8 137 30 103)(9 138 31 104)(10 139 32 105)(11 140 33 106)(12 121 34 107)(13 122 35 108)(14 123 36 109)(15 124 37 110)(16 125 38 111)(17 126 39 112)(18 127 40 113)(19 128 21 114)(20 129 22 115)(41 99 142 78)(42 100 143 79)(43 81 144 80)(44 82 145 61)(45 83 146 62)(46 84 147 63)(47 85 148 64)(48 86 149 65)(49 87 150 66)(50 88 151 67)(51 89 152 68)(52 90 153 69)(53 91 154 70)(54 92 155 71)(55 93 156 72)(56 94 157 73)(57 95 158 74)(58 96 159 75)(59 97 160 76)(60 98 141 77)
(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 79 106 52 23 100 140 153)(2 51 121 78 24 152 107 99)(3 77 108 50 25 98 122 151)(4 49 123 76 26 150 109 97)(5 75 110 48 27 96 124 149)(6 47 125 74 28 148 111 95)(7 73 112 46 29 94 126 147)(8 45 127 72 30 146 113 93)(9 71 114 44 31 92 128 145)(10 43 129 70 32 144 115 91)(11 69 116 42 33 90 130 143)(12 41 131 68 34 142 117 89)(13 67 118 60 35 88 132 141)(14 59 133 66 36 160 119 87)(15 65 120 58 37 86 134 159)(16 57 135 64 38 158 101 85)(17 63 102 56 39 84 136 157)(18 55 137 62 40 156 103 83)(19 61 104 54 21 82 138 155)(20 53 139 80 22 154 105 81)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20AB
order12222244444444444455888810···101010101020···2020···20
size111122222244442020202022202020202···244442···24···4

62 irreducible representations

dim11111122222222244
type+++++++++++--
imageC1C2C2C2C2C4D4D4D5D10D10C4×D5D20C5⋊D4C5⋊D4C8.C22D4.9D10
kernelC4.(C2×D20)C10.Q16C2×C4.Dic5C5×C42⋊C2C22×Dic10C2×Dic10C2×C20C22×C10C42⋊C2C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C10C2
# reps14111831242884428

Matrix representation of C4.(C2×D20) in GL6(𝔽41)

100000
010000
00392800
0013200
0026183013
008291911
,
100000
010000
0021300
00283900
00003013
00001911
,
9300000
11140000
0040121718
002930320
0019151329
00411440
,
4000000
3410000
001336140
003228400
00131149
0020392237

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,39,13,26,8,0,0,28,2,18,29,0,0,0,0,30,19,0,0,0,0,13,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,30,19,0,0,0,0,13,11],[9,11,0,0,0,0,30,14,0,0,0,0,0,0,40,29,19,4,0,0,12,30,15,1,0,0,17,3,13,14,0,0,18,20,29,40],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,13,32,13,20,0,0,36,28,11,39,0,0,1,40,4,22,0,0,40,0,9,37] >;

C4.(C2×D20) in GAP, Magma, Sage, TeX

C_4.(C_2\times D_{20})
% in TeX

G:=Group("C4.(C2xD20)");
// GroupNames label

G:=SmallGroup(320,631);
// by ID

G=gap.SmallGroup(320,631);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,387,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^2=c*a*c^-1=a^-1,a*b=b*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a*c^-1>;
// generators/relations

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