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G = C207D8order 320 = 26·5

1st semidirect product of C20 and D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C207D8, D41D20, C42.49D10, (C5×D4)⋊8D4, (C4×D4)⋊3D5, C43(D4⋊D5), (D4×C20)⋊3C2, C53(C4⋊D8), C204D48C2, C203C823C2, C10.52(C2×D8), C4.13(C2×D20), C20.17(C2×D4), (C2×C20).60D4, C4⋊C4.243D10, C4.9(C4○D20), D206C430C2, (C2×D4).190D10, C20.50(C4○D4), (C4×C20).86C22, C2.8(D4⋊D10), C2.12(C207D4), C10.64(C4⋊D4), (C2×C20).337C23, (C2×D20).97C22, C10.109(C8⋊C22), (D4×C10).232C22, (C2×D4⋊D5)⋊7C2, C2.7(C2×D4⋊D5), (C2×C10).468(C2×D4), (C2×C4).245(C5⋊D4), (C5×C4⋊C4).274C22, (C2×C52C8).93C22, (C2×C4).437(C22×D5), C22.149(C2×C5⋊D4), SmallGroup(320,642)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C207D8
C1C5C10C20C2×C20C2×D20C204D4 — C207D8
C5C10C2×C20 — C207D8
C1C22C42C4×D4

Generators and relations for C207D8
 G = < a,b,c | a20=b8=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 694 in 140 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4 [×9], C23 [×3], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4, C2×D4 [×4], C20 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2×C10 [×4], D4⋊C4 [×2], C4⋊C8, C4×D4, C41D4, C2×D8 [×2], C52C8 [×2], D20 [×8], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×D5 [×2], C22×C10, C4⋊D8, C2×C52C8 [×2], D4⋊D5 [×4], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×D20 [×2], C2×D20 [×2], C22×C20, D4×C10, C203C8, D206C4 [×2], C204D4, C2×D4⋊D5 [×2], D4×C20, C207D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C8⋊C22, D20 [×2], C5⋊D4 [×2], C22×D5, C4⋊D8, D4⋊D5 [×2], C2×D20, C4○D20, C2×C5⋊D4, C207D4, C2×D4⋊D5, D4⋊D10, C207D8

Smallest permutation representation of C207D8
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)
(1 157 130 41 24 74 109 99)(2 156 131 60 25 73 110 98)(3 155 132 59 26 72 111 97)(4 154 133 58 27 71 112 96)(5 153 134 57 28 70 113 95)(6 152 135 56 29 69 114 94)(7 151 136 55 30 68 115 93)(8 150 137 54 31 67 116 92)(9 149 138 53 32 66 117 91)(10 148 139 52 33 65 118 90)(11 147 140 51 34 64 119 89)(12 146 121 50 35 63 120 88)(13 145 122 49 36 62 101 87)(14 144 123 48 37 61 102 86)(15 143 124 47 38 80 103 85)(16 142 125 46 39 79 104 84)(17 141 126 45 40 78 105 83)(18 160 127 44 21 77 106 82)(19 159 128 43 22 76 107 81)(20 158 129 42 23 75 108 100)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(81 155)(82 154)(83 153)(84 152)(85 151)(86 150)(87 149)(88 148)(89 147)(90 146)(91 145)(92 144)(93 143)(94 142)(95 141)(96 160)(97 159)(98 158)(99 157)(100 156)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 140)(120 139)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order12222222444444455888810···1010···1020···2020···20
size1111444040222244422202020202···24···42···24···4

59 irreducible representations

dim11111122222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D8C4○D4D10D10D10C5⋊D4D20C4○D20C8⋊C22D4⋊D5D4⋊D10
kernelC207D8C203C8D206C4C204D4C2×D4⋊D5D4×C20C2×C20C5×D4C4×D4C20C20C42C4⋊C4C2×D4C2×C4D4C4C10C4C2
# reps11212122242222888144

Matrix representation of C207D8 in GL6(𝔽41)

40390000
110000
0064000
001000
0000400
0000040
,
100000
40400000
00211800
00212000
00001229
00001212
,
100000
40400000
0064000
00353500
000010
0000040

G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,21,21,0,0,0,0,18,20,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,40,35,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

C207D8 in GAP, Magma, Sage, TeX

C_{20}\rtimes_7D_8
% in TeX

G:=Group("C20:7D8");
// GroupNames label

G:=SmallGroup(320,642);
// by ID

G=gap.SmallGroup(320,642);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,1123,297,136,12550]);
// Polycyclic

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

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