Copied to
clipboard

G = C10.12- 1+4order 320 = 26·5

1st non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.12- 1+4, C10.72+ 1+4, C20⋊Q88C2, (C2×C20)⋊3Q8, (C2×C4)⋊3Dic10, C20.67(C2×Q8), C4⋊C4.261D10, C4.Dic108C2, C10.9(C22×Q8), (C2×C10).44C24, C4.32(C2×Dic10), (C2×C20).135C23, C20.48D4.6C2, (C22×C4).175D10, C2.11(D46D10), C22.82(C23×D5), C4⋊Dic5.358C22, (C22×C20).74C22, (C2×Dic5).14C23, (C4×Dic5).70C22, C2.11(C22×Dic10), C22.11(C2×Dic10), C10.D4.1C22, C23.223(C22×D5), C23.D5.86C22, (C22×C10).393C23, C2.5(Q8.10D10), C51(C23.41C23), (C2×Dic10).23C22, C23.21D10.20C2, (C2×C4⋊C4).27D5, (C10×C4⋊C4).20C2, (C2×C10).52(C2×Q8), (C5×C4⋊C4).293C22, (C2×C4).570(C22×D5), SmallGroup(320,1172)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.12- 1+4
C1C5C10C2×C10C2×Dic5C4×Dic5C23.21D10 — C10.12- 1+4
C5C2×C10 — C10.12- 1+4
C1C22C2×C4⋊C4

Generators and relations for C10.12- 1+4
 G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a5b, be=eb, dcd-1=ece-1=a5c, ede-1=b2d >

Subgroups: 606 in 206 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C5, C2×C4 [×10], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], Dic10 [×4], C2×Dic5 [×8], C2×C20 [×10], C2×C20 [×2], C22×C10, C23.41C23, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×8], C23.D5 [×4], C5×C4⋊C4 [×4], C2×Dic10 [×4], C22×C20, C22×C20 [×2], C20⋊Q8 [×4], C4.Dic10 [×4], C20.48D4 [×4], C23.21D10 [×2], C10×C4⋊C4, C10.12- 1+4
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4, 2- 1+4, Dic10 [×4], C22×D5 [×7], C23.41C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D46D10, Q8.10D10, C10.12- 1+4

Smallest permutation representation of C10.12- 1+4
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)
(1 153 25 20)(2 152 26 19)(3 151 27 18)(4 160 28 17)(5 159 29 16)(6 158 30 15)(7 157 21 14)(8 156 22 13)(9 155 23 12)(10 154 24 11)(31 139 44 141)(32 138 45 150)(33 137 46 149)(34 136 47 148)(35 135 48 147)(36 134 49 146)(37 133 50 145)(38 132 41 144)(39 131 42 143)(40 140 43 142)(51 114 64 126)(52 113 65 125)(53 112 66 124)(54 111 67 123)(55 120 68 122)(56 119 69 121)(57 118 70 130)(58 117 61 129)(59 116 62 128)(60 115 63 127)(71 99 85 101)(72 98 86 110)(73 97 87 109)(74 96 88 108)(75 95 89 107)(76 94 90 106)(77 93 81 105)(78 92 82 104)(79 91 83 103)(80 100 84 102)
(1 37 6 32)(2 38 7 33)(3 39 8 34)(4 40 9 35)(5 31 10 36)(11 134 16 139)(12 135 17 140)(13 136 18 131)(14 137 19 132)(15 138 20 133)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(51 85 56 90)(52 86 57 81)(53 87 58 82)(54 88 59 83)(55 89 60 84)(61 78 66 73)(62 79 67 74)(63 80 68 75)(64 71 69 76)(65 72 70 77)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(101 121 106 126)(102 122 107 127)(103 123 108 128)(104 124 109 129)(105 125 110 130)(141 154 146 159)(142 155 147 160)(143 156 148 151)(144 157 149 152)(145 158 150 153)
(1 93 25 105)(2 92 26 104)(3 91 27 103)(4 100 28 102)(5 99 29 101)(6 98 30 110)(7 97 21 109)(8 96 22 108)(9 95 23 107)(10 94 24 106)(11 71 154 85)(12 80 155 84)(13 79 156 83)(14 78 157 82)(15 77 158 81)(16 76 159 90)(17 75 160 89)(18 74 151 88)(19 73 152 87)(20 72 153 86)(31 114 44 126)(32 113 45 125)(33 112 46 124)(34 111 47 123)(35 120 48 122)(36 119 49 121)(37 118 50 130)(38 117 41 129)(39 116 42 128)(40 115 43 127)(51 134 64 146)(52 133 65 145)(53 132 66 144)(54 131 67 143)(55 140 68 142)(56 139 69 141)(57 138 70 150)(58 137 61 149)(59 136 62 148)(60 135 63 147)
(1 113 6 118)(2 112 7 117)(3 111 8 116)(4 120 9 115)(5 119 10 114)(11 51 16 56)(12 60 17 55)(13 59 18 54)(14 58 19 53)(15 57 20 52)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(61 152 66 157)(62 151 67 156)(63 160 68 155)(64 159 69 154)(65 158 70 153)(71 146 76 141)(72 145 77 150)(73 144 78 149)(74 143 79 148)(75 142 80 147)(81 138 86 133)(82 137 87 132)(83 136 88 131)(84 135 89 140)(85 134 90 139)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4P5A5B10A···10N20A···20X
order122222444444444···45510···1020···20
size1111222222444420···20222···24···4

62 irreducible representations

dim111111222224444
type++++++-+++-+-
imageC1C2C2C2C2C2Q8D5D10D10Dic102+ 1+42- 1+4D46D10Q8.10D10
kernelC10.12- 1+4C20⋊Q8C4.Dic10C20.48D4C23.21D10C10×C4⋊C4C2×C20C2×C4⋊C4C4⋊C4C22×C4C2×C4C10C10C2C2
# reps1444214286161144

Matrix representation of C10.12- 1+4 in GL6(𝔽41)

2500000
36230000
004000
000400
00390310
00230031
,
3350000
2880000
008061
00003419
00421535
003638428
,
900000
37320000
00323900
000900
00012040
0002610
,
1040000
26310000
0012020
001503240
00310290
002440150
,
3350000
2880000
00150039
00290409
00171012
00310026

G:=sub<GL(6,GF(41))| [25,36,0,0,0,0,0,23,0,0,0,0,0,0,4,0,39,23,0,0,0,4,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,8,0,4,36,0,0,0,0,21,38,0,0,6,34,5,4,0,0,1,19,35,28],[9,37,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,39,9,12,26,0,0,0,0,0,1,0,0,0,0,40,0],[10,26,0,0,0,0,4,31,0,0,0,0,0,0,12,15,31,24,0,0,0,0,0,40,0,0,2,32,29,15,0,0,0,40,0,0],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,15,29,17,31,0,0,0,0,1,0,0,0,0,40,0,0,0,0,39,9,12,26] >;

C10.12- 1+4 in GAP, Magma, Sage, TeX

C_{10}._12_-^{1+4}
% in TeX

G:=Group("C10.1ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1172);
// by ID

G=gap.SmallGroup(320,1172);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,758,675,570,80,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=e^2=a^5,d^2=b^2,b*a*b^-1=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e^-1=a^5*c,e*d*e^-1=b^2*d>;
// generators/relations

׿
×
𝔽