Copied to
clipboard

G = C42.19D10order 320 = 26·5

19th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.19D10, C8⋊C410D5, (C2×C4).26D20, (C2×C20).37D4, (C2×C8).159D10, C204D4.4C2, (C4×C20).4C22, C20.6Q82C2, D205C438C2, C2.8(C8⋊D10), C10.5(C8⋊C22), (C2×D20).9C22, C22.98(C2×D20), C4⋊Dic5.9C22, C4.108(C4○D20), C20.224(C4○D4), (C2×C40).313C22, (C2×C20).734C23, C10.8(C4.4D4), C2.13(C4.D20), C51(C42.29C22), (C5×C8⋊C4)⋊19C2, (C2×C10).117(C2×D4), (C2×C4).678(C22×D5), SmallGroup(320,340)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.19D10
C1C5C10C20C2×C20C2×D20C204D4 — C42.19D10
C5C10C2×C20 — C42.19D10
C1C22C42C8⋊C4

Generators and relations for C42.19D10
 G = < a,b,c,d | a4=b4=1, c10=a2b, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=bc9 >

Subgroups: 638 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], D5 [×2], C10, C10 [×2], C42, C4⋊C4 [×4], C2×C8 [×2], C2×D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C8⋊C4, D4⋊C4 [×4], C42.C2, C41D4, C40 [×2], D20 [×8], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C42.29C22, C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C2×C40 [×2], C2×D20 [×2], C2×D20 [×2], D205C4 [×4], C5×C8⋊C4, C20.6Q8, C204D4, C42.19D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22 [×2], D20 [×2], C22×D5, C42.29C22, C2×D20, C4○D20 [×2], C4.D20, C8⋊D10 [×2], C42.19D10

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222244444455888810···1020···2020···2040···40
size11114040224440402244442···22···24···44···4

56 irreducible representations

dim11111222222244
type++++++++++++
imageC1C2C2C2C2D4D5C4○D4D10D10D20C4○D20C8⋊C22C8⋊D10
kernelC42.19D10D205C4C5×C8⋊C4C20.6Q8C204D4C2×C20C8⋊C4C20C42C2×C8C2×C4C4C10C2
# reps141112242481628

Matrix representation of C42.19D10 in GL6(𝔽41)

120000
40400000
0040000
0004000
000010
000001
,
100000
010000
00301300
00191100
00003013
00001911
,
900000
090000
0000035
0000734
0091600
00361400
,
900000
32320000
00001425
00002527
00271600
00161400

G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,19,0,0,0,0,13,11,0,0,0,0,0,0,30,19,0,0,0,0,13,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,36,0,0,0,0,16,14,0,0,0,7,0,0,0,0,35,34,0,0],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,0,0,27,16,0,0,0,0,16,14,0,0,14,25,0,0,0,0,25,27,0,0] >;

C42.19D10 in GAP, Magma, Sage, TeX

C_4^2._{19}D_{10}
% in TeX

G:=Group("C4^2.19D10");
// GroupNames label

G:=SmallGroup(320,340);
// by ID

G=gap.SmallGroup(320,340);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,142,1123,136,12550]);
// Polycyclic

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

׿
×
𝔽