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G = C42.87D10order 320 = 26·5

87th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.87D10, C10.462- 1+4, C4⋊C4.308D10, (C4×Dic10)⋊5C2, (C2×Dic10)⋊29C4, (C4×C20).20C22, (C2×C10).60C24, C10.36(C23×C4), Dic53Q811C2, (C2×C20).581C23, C20.178(C22×C4), C22⋊C4.123D10, Dic10.46(C2×C4), C42⋊C2.10D5, (C22×C4).185D10, C22.25(C23×D5), C4⋊Dic5.396C22, Dic5.14(C22×C4), (C4×Dic5).75C22, C23.149(C22×D5), C23.D5.90C22, C2.1(D4.10D10), (C22×C20).221C22, (C22×C10).130C23, C52(C23.32C23), (C22×Dic10).18C2, (C2×Dic5).202C23, C23.11D10.5C2, (C2×Dic10).291C22, C10.D4.130C22, C23.21D10.21C2, (C22×Dic5).84C22, C4.57(C2×C4×D5), (C2×C4).57(C4×D5), C2.17(D5×C22×C4), C22.25(C2×C4×D5), (C2×C20).302(C2×C4), (C5×C4⋊C4).301C22, (C2×Dic5).38(C2×C4), (C2×C4).268(C22×D5), (C5×C42⋊C2).11C2, (C2×C10).120(C22×C4), (C5×C22⋊C4).133C22, SmallGroup(320,1188)

Series: Derived Chief Lower central Upper central

C1C10 — C42.87D10
C1C5C10C2×C10C2×Dic5C22×Dic5C22×Dic10 — C42.87D10
C5C10 — C42.87D10
C1C22C42⋊C2

Generators and relations for C42.87D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c-1 >

Subgroups: 686 in 266 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×16], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], Q8 [×16], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×10], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic5 [×8], Dic5 [×4], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C42⋊C2 [×5], C4×Q8 [×8], C22×Q8, Dic10 [×16], C2×Dic5 [×16], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.32C23, C4×Dic5 [×10], C10.D4 [×8], C4⋊Dic5 [×2], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×12], C22×Dic5 [×2], C22×C20, C4×Dic10 [×4], C23.11D10 [×4], Dic53Q8 [×4], C23.21D10, C5×C42⋊C2, C22×Dic10, C42.87D10
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], C4×D5 [×4], C22×D5 [×7], C23.32C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D4.10D10 [×2], C42.87D10

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

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4AB5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order1222224···44···45510···101010101020···2020···20
size1111222···210···10222···244442···24···4

74 irreducible representations

dim1111111122222244
type++++++++++++--
imageC1C2C2C2C2C2C2C4D5D10D10D10D10C4×D52- 1+4D4.10D10
kernelC42.87D10C4×Dic10C23.11D10Dic53Q8C23.21D10C5×C42⋊C2C22×Dic10C2×Dic10C42⋊C2C42C22⋊C4C4⋊C4C22×C4C2×C4C10C2
# reps144411116244421628

Matrix representation of C42.87D10 in GL6(𝔽41)

4000000
0400000
0023200
00373900
00001132
0000930
,
900000
090000
00902323
00093638
0000320
0000032
,
060000
3470000
006700
0035000
001910134
002838734
,
26380000
20150000
00132200
00372800
006142219
0078919

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,37,0,0,0,0,32,39,0,0,0,0,0,0,11,9,0,0,0,0,32,30],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,23,36,32,0,0,0,23,38,0,32],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,6,35,19,28,0,0,7,0,10,38,0,0,0,0,1,7,0,0,0,0,34,34],[26,20,0,0,0,0,38,15,0,0,0,0,0,0,13,37,6,7,0,0,22,28,14,8,0,0,0,0,22,9,0,0,0,0,19,19] >;

C42.87D10 in GAP, Magma, Sage, TeX

C_4^2._{87}D_{10}
% in TeX

G:=Group("C4^2.87D10");
// GroupNames label

G:=SmallGroup(320,1188);
// by ID

G=gap.SmallGroup(320,1188);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,570,80,12550]);
// Polycyclic

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

׿
×
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