Copied to
clipboard

G = C42.91D10order 320 = 26·5

91st non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.91D10, C10.492- 1+4, C10.942+ 1+4, (C4×D20)⋊6C2, C4○D2017C4, D2037(C2×C4), C4⋊C4.310D10, (C4×Dic10)⋊8C2, C42⋊C27D5, Dic1034(C2×C4), (C4×C20).23C22, C10.40(C23×C4), (C2×C10).67C24, Dic54D442C2, C2.2(D48D10), C20.149(C22×C4), (C2×C20).489C23, C22⋊C4.127D10, D10.15(C22×C4), (C22×C4).189D10, C22.29(C23×D5), (C2×D20).294C22, C4⋊Dic5.397C22, Dic5.16(C22×C4), C23.155(C22×D5), C2.2(D4.10D10), (C22×C10).137C23, (C22×C20).227C22, C53(C23.33C23), (C4×Dic5).215C22, (C2×Dic5).206C23, (C22×D5).174C23, D10⋊C4.119C22, (C2×Dic10).323C22, C10.D4.132C22, (C22×Dic5).86C22, (C2×C4)⋊7(C4×D5), C4.94(C2×C4×D5), (D5×C4⋊C4)⋊11C2, (C4×D5)⋊2(C2×C4), (C2×C20)⋊26(C2×C4), C5⋊D411(C2×C4), C22.7(C2×C4×D5), C4⋊C47D511C2, C2.21(D5×C22×C4), (C2×C4⋊Dic5)⋊39C2, (C2×C4×D5).67C22, (C5×C42⋊C2)⋊9C2, (C2×C4○D20).18C2, (C5×C4⋊C4).306C22, (C2×C4).273(C22×D5), (C2×C10).124(C22×C4), (C2×C5⋊D4).106C22, (C5×C22⋊C4).137C22, SmallGroup(320,1195)

Series: Derived Chief Lower central Upper central

C1C10 — C42.91D10
C1C5C10C2×C10C22×D5C2×C4×D5C2×C4○D20 — C42.91D10
C5C10 — C42.91D10
C1C22C42⋊C2

Generators and relations for C42.91D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=c9 >

Subgroups: 926 in 294 conjugacy classes, 151 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×10], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10 [×3], C10 [×2], C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×8], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×4], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4 [×3], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], C4×D5 [×4], D20 [×4], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×C10, C23.33C23, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×4], D10⋊C4 [×4], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20, C4×Dic10 [×2], C4×D20 [×2], Dic54D4 [×4], D5×C4⋊C4 [×2], C4⋊C47D5 [×2], C2×C4⋊Dic5, C5×C42⋊C2, C2×C4○D20, C42.91D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ 1+4, 2- 1+4, C4×D5 [×4], C22×D5 [×7], C23.33C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D48D10, D4.10D10, C42.91D10

Smallest permutation representation of C42.91D10
On 160 points
Generators in S160
(1 53 124 32)(2 44 125 23)(3 55 126 34)(4 46 127 25)(5 57 128 36)(6 48 129 27)(7 59 130 38)(8 50 131 29)(9 41 132 40)(10 52 133 31)(11 43 134 22)(12 54 135 33)(13 45 136 24)(14 56 137 35)(15 47 138 26)(16 58 139 37)(17 49 140 28)(18 60 121 39)(19 51 122 30)(20 42 123 21)(61 97 104 144)(62 88 105 155)(63 99 106 146)(64 90 107 157)(65 81 108 148)(66 92 109 159)(67 83 110 150)(68 94 111 141)(69 85 112 152)(70 96 113 143)(71 87 114 154)(72 98 115 145)(73 89 116 156)(74 100 117 147)(75 91 118 158)(76 82 119 149)(77 93 120 160)(78 84 101 151)(79 95 102 142)(80 86 103 153)
(1 88 134 145)(2 99 135 156)(3 90 136 147)(4 81 137 158)(5 92 138 149)(6 83 139 160)(7 94 140 151)(8 85 121 142)(9 96 122 153)(10 87 123 144)(11 98 124 155)(12 89 125 146)(13 100 126 157)(14 91 127 148)(15 82 128 159)(16 93 129 150)(17 84 130 141)(18 95 131 152)(19 86 132 143)(20 97 133 154)(21 61 52 114)(22 72 53 105)(23 63 54 116)(24 74 55 107)(25 65 56 118)(26 76 57 109)(27 67 58 120)(28 78 59 111)(29 69 60 102)(30 80 41 113)(31 71 42 104)(32 62 43 115)(33 73 44 106)(34 64 45 117)(35 75 46 108)(36 66 47 119)(37 77 48 110)(38 68 49 101)(39 79 50 112)(40 70 51 103)
(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)
(1 114 11 104)(2 103 12 113)(3 112 13 102)(4 101 14 111)(5 110 15 120)(6 119 16 109)(7 108 17 118)(8 117 18 107)(9 106 19 116)(10 115 20 105)(21 98 31 88)(22 87 32 97)(23 96 33 86)(24 85 34 95)(25 94 35 84)(26 83 36 93)(27 92 37 82)(28 81 38 91)(29 90 39 100)(30 99 40 89)(41 156 51 146)(42 145 52 155)(43 154 53 144)(44 143 54 153)(45 152 55 142)(46 141 56 151)(47 150 57 160)(48 159 58 149)(49 148 59 158)(50 157 60 147)(61 124 71 134)(62 133 72 123)(63 122 73 132)(64 131 74 121)(65 140 75 130)(66 129 76 139)(67 138 77 128)(68 127 78 137)(69 136 79 126)(70 125 80 135)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4L4M···4X5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order12222222224···44···45510···101010101020···2020···20
size111122101010102···210···10222···244442···24···4

74 irreducible representations

dim11111111112222224444
type+++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C4D5D10D10D10D10C4×D52+ 1+42- 1+4D48D10D4.10D10
kernelC42.91D10C4×Dic10C4×D20Dic54D4D5×C4⋊C4C4⋊C47D5C2×C4⋊Dic5C5×C42⋊C2C2×C4○D20C4○D20C42⋊C2C42C22⋊C4C4⋊C4C22×C4C2×C4C10C10C2C2
# reps1224221111624442161144

Matrix representation of C42.91D10 in GL6(𝔽41)

3200000
0320000
00403434
0004141
00305370
003136037
,
900000
090000
0038301823
0063021
0026343311
00034308
,
760000
3400000
0061131
003503737
001721530
0013341130
,
34400000
770000
001210229
0025291910
0029191017
0031313931

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,4,0,30,31,0,0,0,4,5,36,0,0,34,14,37,0,0,0,34,1,0,37],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,38,6,26,0,0,0,30,3,34,34,0,0,18,0,33,30,0,0,23,21,11,8],[7,34,0,0,0,0,6,0,0,0,0,0,0,0,6,35,17,13,0,0,11,0,21,34,0,0,3,37,5,11,0,0,1,37,30,30],[34,7,0,0,0,0,40,7,0,0,0,0,0,0,12,25,29,31,0,0,10,29,19,31,0,0,22,19,10,39,0,0,9,10,17,31] >;

C42.91D10 in GAP, Magma, Sage, TeX

C_4^2._{91}D_{10}
% in TeX

G:=Group("C4^2.91D10");
// GroupNames label

G:=SmallGroup(320,1195);
// by ID

G=gap.SmallGroup(320,1195);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,570,297,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations

׿
×
𝔽