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G = C4011D4order 320 = 26·5

11st semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4011D4, (C2×D8)⋊8D5, C52C89D4, (C10×D8)⋊9C2, C83(C5⋊D4), C54(C83D4), C408C49C2, C4.21(D4×D5), C20⋊D45C2, (C2×C8).85D10, (C2×D4).62D10, C20.164(C2×D4), C20.17D44C2, (C2×Dic5).72D4, C22.254(D4×D5), C2.28(D8⋊D5), C2.18(C20⋊D4), C10.27(C41D4), C10.49(C8⋊C22), (C2×C20).431C23, (C2×C40).147C22, (D4×C10).81C22, (C2×D20).119C22, (C4×Dic5).51C22, (C2×Dic10).124C22, C4.5(C2×C5⋊D4), (C2×D4⋊D5)⋊18C2, (C2×C40⋊C2)⋊23C2, (C2×D4.D5)⋊17C2, (C2×C10).344(C2×D4), (C2×C4).521(C22×D5), (C2×C52C8).148C22, SmallGroup(320,781)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4011D4
C1C5C10C20C2×C20C4×Dic5C20⋊D4 — C4011D4
C5C10C2×C20 — C4011D4
C1C22C2×C4C2×D8

Generators and relations for C4011D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a19, cbc=b-1 >

Subgroups: 686 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×3], D4 [×10], Q8 [×2], C23 [×3], D5, C10, C10 [×2], C10 [×2], C42, C22⋊C4 [×2], C2×C8, C2×C8, D8 [×4], SD16 [×4], C2×D4 [×2], C2×D4 [×3], C2×Q8, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×6], C8⋊C4, C4.4D4, C41D4, C2×D8, C2×D8, C2×SD16 [×2], C52C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C83D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, D4⋊D5 [×2], D4.D5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×2], C2×Dic10, C2×D20, C2×C5⋊D4 [×2], D4×C10 [×2], C408C4, C2×C40⋊C2, C2×D4⋊D5, C2×D4.D5, C20.17D4, C20⋊D4, C10×D8, C4011D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C83D4, D4×D5 [×2], C2×C5⋊D4, D8⋊D5 [×2], C20⋊D4, C4011D4

Smallest permutation representation of C4011D4
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)
(1 107 136 46)(2 96 137 75)(3 85 138 64)(4 114 139 53)(5 103 140 42)(6 92 141 71)(7 81 142 60)(8 110 143 49)(9 99 144 78)(10 88 145 67)(11 117 146 56)(12 106 147 45)(13 95 148 74)(14 84 149 63)(15 113 150 52)(16 102 151 41)(17 91 152 70)(18 120 153 59)(19 109 154 48)(20 98 155 77)(21 87 156 66)(22 116 157 55)(23 105 158 44)(24 94 159 73)(25 83 160 62)(26 112 121 51)(27 101 122 80)(28 90 123 69)(29 119 124 58)(30 108 125 47)(31 97 126 76)(32 86 127 65)(33 115 128 54)(34 104 129 43)(35 93 130 72)(36 82 131 61)(37 111 132 50)(38 100 133 79)(39 89 134 68)(40 118 135 57)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 92)(42 111)(43 90)(44 109)(45 88)(46 107)(47 86)(48 105)(49 84)(50 103)(51 82)(52 101)(53 120)(54 99)(55 118)(56 97)(57 116)(58 95)(59 114)(60 93)(61 112)(62 91)(63 110)(64 89)(65 108)(66 87)(67 106)(68 85)(69 104)(70 83)(71 102)(72 81)(73 100)(74 119)(75 98)(76 117)(77 96)(78 115)(79 94)(80 113)(121 131)(122 150)(123 129)(124 148)(125 127)(126 146)(128 144)(130 142)(132 140)(133 159)(134 138)(135 157)(137 155)(139 153)(141 151)(143 149)(145 147)(152 160)(154 158)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D40A···40H
order12222224444455888810···1010···102020202040···40
size1111884022202040224420202···28···844444···4

44 irreducible representations

dim1111111122222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5D10D10C5⋊D4C8⋊C22D4×D5D4×D5D8⋊D5
kernelC4011D4C408C4C2×C40⋊C2C2×D4⋊D5C2×D4.D5C20.17D4C20⋊D4C10×D8C52C8C40C2×Dic5C2×D8C2×C8C2×D4C8C10C4C22C2
# reps1111111122222482228

Matrix representation of C4011D4 in GL10(𝔽41)

1000000000
0100000000
0020382040000
0026352000000
0021372130000
002101560000
00000035381111
0000003111247
00000033620
0000006131834
,
17200000000
192400000000
00212420240000
00212026210000
00202421240000
00262121200000
000000341190
000000253322
0000003732034
000000743915
,
40000000000
17100000000
0061000000
00635000000
000035400000
00003560000
0000001000
000000234000
00000022710
00000026174040

G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20,26,21,21,0,0,0,0,0,0,38,35,37,0,0,0,0,0,0,0,20,20,21,15,0,0,0,0,0,0,4,0,3,6,0,0,0,0,0,0,0,0,0,0,35,31,3,6,0,0,0,0,0,0,38,11,36,13,0,0,0,0,0,0,11,24,2,18,0,0,0,0,0,0,11,7,0,34],[17,19,0,0,0,0,0,0,0,0,2,24,0,0,0,0,0,0,0,0,0,0,21,21,20,26,0,0,0,0,0,0,24,20,24,21,0,0,0,0,0,0,20,26,21,21,0,0,0,0,0,0,24,21,24,20,0,0,0,0,0,0,0,0,0,0,34,25,37,7,0,0,0,0,0,0,11,33,32,4,0,0,0,0,0,0,9,2,0,39,0,0,0,0,0,0,0,2,34,15],[40,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,1,23,22,26,0,0,0,0,0,0,0,40,7,17,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,40] >;

C4011D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{11}D_4
% in TeX

G:=Group("C40:11D4");
// GroupNames label

G:=SmallGroup(320,781);
// by ID

G=gap.SmallGroup(320,781);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,1094,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations

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