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G = C42.114D10order 320 = 26·5

114th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.114D10, C10.202+ 1+4, (C4×D4)⋊21D5, (D4×C20)⋊23C2, C4⋊C4.319D10, C20⋊D4.9C2, D208C416C2, (C4×Dic10)⋊34C2, (C2×D4).220D10, C4.16(C4○D20), C4.D2019C2, C20.17D49C2, (C22×C4).48D10, Dic53Q816C2, D10.12D48C2, C20.111(C4○D4), (C4×C20).158C22, (C2×C20).701C23, (C2×C10).103C24, C22⋊C4.116D10, Dic5.5D48C2, C2.21(D46D10), Dic5.61(C4○D4), (C2×D20).145C22, (D4×C10).263C22, C4⋊Dic5.301C22, (C2×Dic5).44C23, (C4×Dic5).84C22, (C22×D5).37C23, C22.128(C23×D5), C23.100(C22×D5), D10⋊C4.87C22, C23.23D1018C2, (C22×C20).365C22, (C22×C10).173C23, C51(C22.53C24), C10.D4.66C22, C23.D5.107C22, (C2×Dic10).151C22, (C4×C5⋊D4)⋊45C2, C2.26(D5×C4○D4), C10.45(C2×C4○D4), C2.52(C2×C4○D20), (C2×C4×D5).253C22, (C5×C4⋊C4).332C22, (C2×C4).286(C22×D5), (C2×C5⋊D4).124C22, (C5×C22⋊C4).127C22, SmallGroup(320,1231)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.114D10
C1C5C10C2×C10C22×D5C2×C4×D5D10.12D4 — C42.114D10
C5C2×C10 — C42.114D10
C1C22C4×D4

Generators and relations for C42.114D10
 G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 838 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×6], C4×D4, C4×D4 [×3], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C41D4, Dic10 [×4], C4×D5 [×2], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.53C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C4.D20, D10.12D4 [×2], Dic5.5D4 [×2], Dic53Q8, D208C4, C4×C5⋊D4 [×2], C23.23D10 [×2], C20.17D4, C20⋊D4, D4×C20, C42.114D10
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], C23×D5, C2×C4○D20, D46D10, D5×C4○D4, C42.114D10

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

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L4M4N4O4P4Q5A5B10A···10F10G···10N20A···20H20I···20X
order122222224···44444444445510···1010···1020···2020···20
size11114420202···241010101020202020222···24···42···24···4

65 irreducible representations

dim111111111111222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D10D10C4○D202+ 1+4D46D10D5×C4○D4
kernelC42.114D10C4×Dic10C4.D20D10.12D4Dic5.5D4Dic53Q8D208C4C4×C5⋊D4C23.23D10C20.17D4C20⋊D4D4×C20C4×D4Dic5C20C42C22⋊C4C4⋊C4C22×C4C2×D4C4C10C2C2
# reps1112211221112442424216144

Matrix representation of C42.114D10 in GL6(𝔽41)

3240000
090000
001000
000100
00002629
0000515
,
9370000
0320000
0040000
0004000
0000320
0000032
,
900000
090000
00343400
007100
0000913
0000032
,
3200000
2190000
00343400
001700
000090
000009

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,4,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,5,0,0,0,0,29,15],[9,0,0,0,0,0,37,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,9,0,0,0,0,0,13,32],[32,21,0,0,0,0,0,9,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C42.114D10 in GAP, Magma, Sage, TeX

C_4^2._{114}D_{10}
% in TeX

G:=Group("C4^2.114D10");
// GroupNames label

G:=SmallGroup(320,1231);
// by ID

G=gap.SmallGroup(320,1231);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,136,12550]);
// Polycyclic

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

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