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G = C8.8D20order 320 = 26·5

4th non-split extension by C8 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.8D20, C40.58D4, C42.263D10, (C4×C8)⋊9D5, (C4×C40)⋊14C2, (C2×D40).2C2, C4.32(C2×D20), (C2×C4).62D20, (C2×Dic20)⋊1C2, C10.4(C4○D8), (C2×C20).352D4, C20.275(C2×D4), (C2×C8).318D10, C4.D201C2, C51(C8.12D4), C2.7(C204D4), C10.5(C41D4), (C2×D20).6C22, C22.92(C2×D20), C2.7(D407C2), (C2×C40).390C22, (C4×C20).309C22, (C2×C20).725C23, (C2×Dic10).5C22, (C2×C40⋊C2)⋊8C2, (C2×C10).108(C2×D4), (C2×C4).668(C22×D5), SmallGroup(320,323)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8.8D20
C1C5C10C20C2×C20C2×D20C4.D20 — C8.8D20
C5C10C2×C20 — C8.8D20
C1C22C42C4×C8

Generators and relations for C8.8D20
 G = < a,b,c | a8=b20=1, c2=a4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 686 in 130 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C40 [×4], Dic10 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C8.12D4, C40⋊C2 [×4], D40 [×2], Dic20 [×2], D10⋊C4 [×4], C4×C20, C2×C40 [×2], C2×Dic10 [×2], C2×D20 [×2], C4×C40, C4.D20 [×2], C2×C40⋊C2 [×2], C2×D40, C2×Dic20, C8.8D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C4○D8 [×2], D20 [×6], C22×D5, C8.12D4, C2×D20 [×3], C204D4, D407C2 [×2], C8.8D20

Smallest permutation representation of C8.8D20
On 160 points
Generators in S160
(1 122 85 26 141 59 102 71)(2 123 86 27 142 60 103 72)(3 124 87 28 143 41 104 73)(4 125 88 29 144 42 105 74)(5 126 89 30 145 43 106 75)(6 127 90 31 146 44 107 76)(7 128 91 32 147 45 108 77)(8 129 92 33 148 46 109 78)(9 130 93 34 149 47 110 79)(10 131 94 35 150 48 111 80)(11 132 95 36 151 49 112 61)(12 133 96 37 152 50 113 62)(13 134 97 38 153 51 114 63)(14 135 98 39 154 52 115 64)(15 136 99 40 155 53 116 65)(16 137 100 21 156 54 117 66)(17 138 81 22 157 55 118 67)(18 139 82 23 158 56 119 68)(19 140 83 24 159 57 120 69)(20 121 84 25 160 58 101 70)
(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)
(1 160 141 20)(2 19 142 159)(3 158 143 18)(4 17 144 157)(5 156 145 16)(6 15 146 155)(7 154 147 14)(8 13 148 153)(9 152 149 12)(10 11 150 151)(21 43 66 126)(22 125 67 42)(23 41 68 124)(24 123 69 60)(25 59 70 122)(26 121 71 58)(27 57 72 140)(28 139 73 56)(29 55 74 138)(30 137 75 54)(31 53 76 136)(32 135 77 52)(33 51 78 134)(34 133 79 50)(35 49 80 132)(36 131 61 48)(37 47 62 130)(38 129 63 46)(39 45 64 128)(40 127 65 44)(81 88 118 105)(82 104 119 87)(83 86 120 103)(84 102 101 85)(89 100 106 117)(90 116 107 99)(91 98 108 115)(92 114 109 97)(93 96 110 113)(94 112 111 95)

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

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

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

86 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B8A···8H10A···10F20A···20X40A···40AF
order1222224···444558···810···1020···2040···40
size111140402···24040222···22···22···22···2

86 irreducible representations

dim111111222222222
type+++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10C4○D8D20D20D407C2
kernelC8.8D20C4×C40C4.D20C2×C40⋊C2C2×D40C2×Dic20C40C2×C20C4×C8C42C2×C8C10C8C2×C4C2
# reps11221142224816832

Matrix representation of C8.8D20 in GL4(𝔽41) generated by

12800
33500
00400
00040
,
13900
32000
003230
001127
,
91300
03200
003230
00119
G:=sub<GL(4,GF(41))| [12,33,0,0,8,5,0,0,0,0,40,0,0,0,0,40],[13,32,0,0,9,0,0,0,0,0,32,11,0,0,30,27],[9,0,0,0,13,32,0,0,0,0,32,11,0,0,30,9] >;

C8.8D20 in GAP, Magma, Sage, TeX

C_8._8D_{20}
% in TeX

G:=Group("C8.8D20");
// GroupNames label

G:=SmallGroup(320,323);
// by ID

G=gap.SmallGroup(320,323);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=a^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations

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