Copied to
clipboard

G = C4012D4order 320 = 26·5

12nd semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4012D4, (C2×D8)⋊9D5, C84(C5⋊D4), C55(C82D4), (C10×D8)⋊10C2, C202D46C2, C406C421C2, (C2×C8).86D10, (C2×D4).66D10, C20.168(C2×D4), C20.94(C4○D4), D4⋊Dic531C2, (C2×Dic5).74D4, (C22×D5).44D4, C22.259(D4×D5), C4.29(D42D5), C2.31(D8⋊D5), C2.17(C202D4), C10.52(C8⋊C22), (C2×C40).148C22, (C2×C20).436C23, (D4×C10).85C22, C10.110(C4⋊D4), C4⋊Dic5.167C22, (C2×C8⋊D5)⋊7C2, C4.80(C2×C5⋊D4), (C2×C4×D5).50C22, (C2×C10).349(C2×D4), (C2×C4).526(C22×D5), (C2×C52C8).150C22, SmallGroup(320,786)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4012D4
C1C5C10C20C2×C20C2×C4×D5C202D4 — C4012D4
C5C10C2×C20 — C4012D4
C1C22C2×C4C2×D8

Generators and relations for C4012D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a29, cbc=b-1 >

Subgroups: 566 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8, C2×C4, C2×C4 [×5], D4 [×8], C23 [×3], D5, C10, C10 [×2], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], D8 [×2], C22×C4, C2×D4 [×2], C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×6], D4⋊C4 [×2], C4.Q8, C4⋊D4 [×2], C2×M4(2), C2×D8, C52C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C82D4, C8⋊D5 [×2], C2×C52C8, C4⋊Dic5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10 [×2], C406C4, D4⋊Dic5 [×2], C2×C8⋊D5, C202D4 [×2], C10×D8, C4012D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C82D4, D4×D5, D42D5, C2×C5⋊D4, D8⋊D5 [×2], C202D4, C4012D4

Smallest permutation representation of C4012D4
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 66 152 96)(2 45 153 115)(3 64 154 94)(4 43 155 113)(5 62 156 92)(6 41 157 111)(7 60 158 90)(8 79 159 109)(9 58 160 88)(10 77 121 107)(11 56 122 86)(12 75 123 105)(13 54 124 84)(14 73 125 103)(15 52 126 82)(16 71 127 101)(17 50 128 120)(18 69 129 99)(19 48 130 118)(20 67 131 97)(21 46 132 116)(22 65 133 95)(23 44 134 114)(24 63 135 93)(25 42 136 112)(26 61 137 91)(27 80 138 110)(28 59 139 89)(29 78 140 108)(30 57 141 87)(31 76 142 106)(32 55 143 85)(33 74 144 104)(34 53 145 83)(35 72 146 102)(36 51 147 81)(37 70 148 100)(38 49 149 119)(39 68 150 98)(40 47 151 117)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 91)(42 120)(43 109)(44 98)(45 87)(46 116)(47 105)(48 94)(49 83)(50 112)(51 101)(52 90)(53 119)(54 108)(55 97)(56 86)(57 115)(58 104)(59 93)(60 82)(61 111)(62 100)(63 89)(64 118)(65 107)(66 96)(67 85)(68 114)(69 103)(70 92)(71 81)(72 110)(73 99)(74 88)(75 117)(76 106)(77 95)(78 84)(79 113)(80 102)(121 133)(123 151)(124 140)(125 129)(126 158)(127 147)(128 136)(130 154)(131 143)(134 150)(135 139)(137 157)(138 146)(141 153)(144 160)(145 149)(148 156)(155 159)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D40A···40H
order12222224444455888810···1010···102020202040···40
size1111882022204040224420202···28···844444···4

44 irreducible representations

dim111111222222224444
type+++++++++++++-+
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10C5⋊D4C8⋊C22D42D5D4×D5D8⋊D5
kernelC4012D4C406C4D4⋊Dic5C2×C8⋊D5C202D4C10×D8C40C2×Dic5C22×D5C2×D8C20C2×C8C2×D4C8C10C4C22C2
# reps112121211222482228

Matrix representation of C4012D4 in GL6(𝔽41)

4000000
0400000
0031900
0032900
004421
0020057
,
010000
4000000
00332733
00182320
003611740
0016304034
,
4000000
010000
001000
00344000
00132267
009373635

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,31,32,4,20,0,0,9,9,4,0,0,0,0,0,2,5,0,0,0,0,1,7],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,33,1,36,16,0,0,27,8,11,30,0,0,3,23,7,40,0,0,3,20,40,34],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,34,13,9,0,0,0,40,22,37,0,0,0,0,6,36,0,0,0,0,7,35] >;

C4012D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{12}D_4
% in TeX

G:=Group("C40:12D4");
// GroupNames label

G:=SmallGroup(320,786);
// by ID

G=gap.SmallGroup(320,786);
# by ID

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

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

׿
×
𝔽