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G = C42.136D10order 320 = 26·5

136th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.136D10, C10.1132+ 1+4, (C4×Q8)⋊18D5, (C4×D20)⋊42C2, (Q8×C20)⋊20C2, C4⋊C4.303D10, D208C418C2, (C4×Dic10)⋊42C2, C4.19(C4○D20), C204D4.11C2, C4.D2021C2, (C2×Q8).184D10, C20.123(C4○D4), C20.23D410C2, (C2×C20).592C23, (C2×C10).129C24, (C4×C20).181C22, C4.51(Q82D5), (C2×D20).31C22, D10.13D410C2, C2.25(D48D10), C4⋊Dic5.401C22, (Q8×C10).229C22, (C4×Dic5).96C22, (C2×Dic5).59C23, (C22×D5).51C23, C22.150(C23×D5), C52(C22.53C24), D10⋊C4.145C22, (C2×Dic10).252C22, C10.D4.116C22, C10.58(C2×C4○D4), C2.68(C2×C4○D20), C2.14(C2×Q82D5), (C2×C4×D5).258C22, (C5×C4⋊C4).357C22, (C2×C4).291(C22×D5), SmallGroup(320,1257)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.136D10
C1C5C10C2×C10C22×D5C2×C4×D5D10.13D4 — C42.136D10
C5C2×C10 — C42.136D10
C1C22C4×Q8

Generators and relations for C42.136D10
 G = < a,b,c,d | a4=b4=d2=1, c10=a2, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=b-1, dcd=a2c9 >

Subgroups: 934 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4 [×10], Q8 [×4], C23 [×4], D5 [×4], C10 [×3], C42, C42 [×2], C42 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×6], C2×Q8, C2×Q8, Dic5 [×4], C20 [×4], C20 [×5], D10 [×12], C2×C10, C4×D4 [×4], C4×Q8, C4×Q8, C22.D4 [×4], C4.4D4 [×4], C41D4, Dic10 [×2], C4×D5 [×4], D20 [×10], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×4], C22.53C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×12], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×4], C2×D20 [×6], Q8×C10, C4×Dic10, C4×D20 [×2], C204D4, C4.D20 [×2], D208C4 [×2], D10.13D4 [×4], C20.23D4 [×2], Q8×C20, C42.136D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.53C24, C4○D20 [×2], Q82D5 [×2], C23×D5, C2×C4○D20, C2×Q82D5, D48D10, C42.136D10

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

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L4M4N4O4P4Q5A5B10A···10F20A···20H20I···20AF
order122222224···44444444445510···1020···2020···20
size1111202020202···2444101010102020222···22···24···4

65 irreducible representations

dim111111111222222444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2D5C4○D4D10D10D10C4○D202+ 1+4Q82D5D48D10
kernelC42.136D10C4×Dic10C4×D20C204D4C4.D20D208C4D10.13D4C20.23D4Q8×C20C4×Q8C20C42C4⋊C4C2×Q8C4C10C4C2
# reps1121224212866216144

Matrix representation of C42.136D10 in GL4(𝔽41) generated by

9000
0900
00923
00032
,
2400
93900
00400
00040
,
251200
271300
00139
00140
,
6500
343500
0010
00140
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,23,32],[2,9,0,0,4,39,0,0,0,0,40,0,0,0,0,40],[25,27,0,0,12,13,0,0,0,0,1,1,0,0,39,40],[6,34,0,0,5,35,0,0,0,0,1,1,0,0,0,40] >;

C42.136D10 in GAP, Magma, Sage, TeX

C_4^2._{136}D_{10}
% in TeX

G:=Group("C4^2.136D10");
// GroupNames label

G:=SmallGroup(320,1257);
// by ID

G=gap.SmallGroup(320,1257);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,184,1571,192,12550]);
// Polycyclic

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

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