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G = D204C8order 320 = 26·5

2nd semidirect product of D20 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D204C8, C20.54D8, C20.49SD16, C42.192D10, C20.17M4(2), C4⋊C81D5, C54(D4⋊C8), C4.1(C8×D5), C10.22C4≀C2, C20.27(C2×C8), (C4×D20).6C2, C4.27(D4⋊D5), (C2×D20).23C4, (C2×C20).225D4, (C2×C4).109D20, C4.1(C8⋊D5), C4.15(Q8⋊D5), C4⋊Dic5.26C4, (C4×C20).41C22, C2.1(D206C4), C2.7(D101C8), C2.1(D207C4), C10.20(C22⋊C8), C10.18(D4⋊C4), C22.34(D10⋊C4), (C5×C4⋊C8)⋊1C2, (C4×C52C8)⋊1C2, (C2×C4).65(C4×D5), (C2×C20).221(C2×C4), (C2×C4).265(C5⋊D4), (C2×C10).109(C22⋊C4), SmallGroup(320,41)

Series: Derived Chief Lower central Upper central

C1C20 — D204C8
C1C5C10C2×C10C2×C20C4×C20C4×D20 — D204C8
C5C10C20 — D204C8
C1C2×C4C42C4⋊C8

Generators and relations for D204C8
 G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a15b >

Subgroups: 374 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×3], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, Dic5, C20 [×4], C20, D10 [×4], C2×C10, C4×C8, C4⋊C8, C4×D4, C52C8 [×2], C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C22×D5, D4⋊C8, C2×C52C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C4×C52C8, C5×C4⋊C8, C4×D20, D204C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D8, SD16, D10, C22⋊C8, D4⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, D4⋊C8, C8×D5, C8⋊D5, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, D101C8, D207C4, D204C8

Smallest permutation representation of D204C8
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)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 140)(10 139)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 130)(20 129)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 60)(40 59)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 100)(80 99)(101 151)(102 150)(103 149)(104 148)(105 147)(106 146)(107 145)(108 144)(109 143)(110 142)(111 141)(112 160)(113 159)(114 158)(115 157)(116 156)(117 155)(118 154)(119 153)(120 152)
(1 42 107 77 129 33 146 98)(2 53 108 68 130 24 147 89)(3 44 109 79 131 35 148 100)(4 55 110 70 132 26 149 91)(5 46 111 61 133 37 150 82)(6 57 112 72 134 28 151 93)(7 48 113 63 135 39 152 84)(8 59 114 74 136 30 153 95)(9 50 115 65 137 21 154 86)(10 41 116 76 138 32 155 97)(11 52 117 67 139 23 156 88)(12 43 118 78 140 34 157 99)(13 54 119 69 121 25 158 90)(14 45 120 80 122 36 159 81)(15 56 101 71 123 27 160 92)(16 47 102 62 124 38 141 83)(17 58 103 73 125 29 142 94)(18 49 104 64 126 40 143 85)(19 60 105 75 127 31 144 96)(20 51 106 66 128 22 145 87)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E···8L10A···10F20A···20H20I···20P40A···40P
order12222244444444445588888···810···1020···2020···2040···40
size1111202011112222202022444410···102···22···24···44···4

68 irreducible representations

dim1111111222222222222444
type+++++++++++
imageC1C2C2C2C4C4C8D4D5M4(2)D8SD16D10C4≀C2C4×D5D20C5⋊D4C8×D5C8⋊D5D4⋊D5Q8⋊D5D207C4
kernelD204C8C4×C52C8C5×C4⋊C8C4×D20C4⋊Dic5C2×D20D20C2×C20C4⋊C8C20C20C20C42C10C2×C4C2×C4C2×C4C4C4C4C4C2
# reps1111228222222444488224

Matrix representation of D204C8 in GL6(𝔽41)

4000000
0400000
0004000
0013500
000001
0000400
,
100000
21400000
0004000
0040000
000001
000010
,
21390000
20200000
00392800
0013200
00002626
00002615

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[21,20,0,0,0,0,39,20,0,0,0,0,0,0,39,13,0,0,0,0,28,2,0,0,0,0,0,0,26,26,0,0,0,0,26,15] >;

D204C8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_4C_8
% in TeX

G:=Group("D20:4C8");
// GroupNames label

G:=SmallGroup(320,41);
// by ID

G=gap.SmallGroup(320,41);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,100,1123,570,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

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