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G = D105M4(2)  order 320 = 26·5

3rd semidirect product of D10 and M4(2) acting via M4(2)/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D105M4(2), C42.29D10, C4⋊C812D5, C52C827D4, C58(C89D4), (C4×D20).8C2, C10.77(C4×D4), C4.206(D4×D5), (C2×D20).20C4, C20.365(C2×D4), (C2×C8).217D10, C4⋊Dic5.24C4, D101C823C2, C10.36(C8○D4), (C4×C20).62C22, C42.D53C2, C2.17(D5×M4(2)), D10⋊C4.23C4, C20.335(C4○D4), C2.7(D208C4), (C2×C40).212C22, (C2×C20).833C23, C4.55(Q82D5), C10.62(C2×M4(2)), C2.14(D20.3C4), (D5×C2×C8)⋊23C2, (C5×C4⋊C8)⋊17C2, (C2×C4).35(C4×D5), (C2×C8⋊D5)⋊21C2, C22.111(C2×C4×D5), (C2×C20).216(C2×C4), (C2×C4×D5).349C22, (C2×Dic5).24(C2×C4), (C22×D5).75(C2×C4), (C2×C4).775(C22×D5), (C2×C10).189(C22×C4), (C2×C52C8).199C22, SmallGroup(320,463)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D105M4(2)
C1C5C10C20C2×C20C2×C4×D5C4×D20 — D105M4(2)
C5C2×C10 — D105M4(2)
C1C2×C4C4⋊C8

Generators and relations for D105M4(2)
 G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, ac=ca, cbc-1=a5b, dbd=a3b, dcd=c5 >

Subgroups: 446 in 124 conjugacy classes, 51 normal (47 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×5], C2×C4 [×3], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8 [×2], C52C8, C40 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C89D4, C8×D5 [×2], C8⋊D5 [×2], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20, C42.D5, D101C8 [×2], C5×C4⋊C8, C4×D20, D5×C2×C8, C2×C8⋊D5, D105M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C22×D5, C89D4, C2×C4×D5, D4×D5, Q82D5, D208C4, D20.3C4, D5×M4(2), D105M4(2)

Smallest permutation representation of D105M4(2)
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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 120)(9 119)(10 118)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 70)(21 128)(22 127)(23 126)(24 125)(25 124)(26 123)(27 122)(28 121)(29 130)(30 129)(31 94)(32 93)(33 92)(34 91)(35 100)(36 99)(37 98)(38 97)(39 96)(40 95)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 110)(50 109)(51 159)(52 158)(53 157)(54 156)(55 155)(56 154)(57 153)(58 152)(59 151)(60 160)(71 146)(72 145)(73 144)(74 143)(75 142)(76 141)(77 150)(78 149)(79 148)(80 147)(81 136)(82 135)(83 134)(84 133)(85 132)(86 131)(87 140)(88 139)(89 138)(90 137)
(1 145 45 11 25 138 38 158)(2 146 46 12 26 139 39 159)(3 147 47 13 27 140 40 160)(4 148 48 14 28 131 31 151)(5 149 49 15 29 132 32 152)(6 150 50 16 30 133 33 153)(7 141 41 17 21 134 34 154)(8 142 42 18 22 135 35 155)(9 143 43 19 23 136 36 156)(10 144 44 20 24 137 37 157)(51 111 71 108 68 128 88 91)(52 112 72 109 69 129 89 92)(53 113 73 110 70 130 90 93)(54 114 74 101 61 121 81 94)(55 115 75 102 62 122 82 95)(56 116 76 103 63 123 83 96)(57 117 77 104 64 124 84 97)(58 118 78 105 65 125 85 98)(59 119 79 106 66 126 86 99)(60 120 80 107 67 127 87 100)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 30)(7 29)(8 28)(9 27)(10 26)(12 20)(13 19)(14 18)(15 17)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 70)(66 69)(67 68)(71 80)(72 79)(73 78)(74 77)(75 76)(81 84)(82 83)(85 90)(86 89)(87 88)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 110)(99 109)(100 108)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 130)(119 129)(120 128)(131 135)(132 134)(136 140)(137 139)(141 149)(142 148)(143 147)(144 146)(151 155)(152 154)(156 160)(157 159)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F20A···20H20I···20P40A···40P
order12222224444444445588888888888810···1020···2020···2040···40
size1111101020111144101020222222441010101020202···22···24···44···4

68 irreducible representations

dim1111111111222222222444
type+++++++++++++
imageC1C2C2C2C2C2C2C4C4C4D4D5C4○D4M4(2)D10D10C8○D4C4×D5D20.3C4D4×D5Q82D5D5×M4(2)
kernelD105M4(2)C42.D5D101C8C5×C4⋊C8C4×D20D5×C2×C8C2×C8⋊D5C4⋊Dic5D10⋊C4C2×D20C52C8C4⋊C8C20D10C42C2×C8C10C2×C4C2C4C4C2
# reps11211112422224244816224

Matrix representation of D105M4(2) in GL6(𝔽41)

4000000
0400000
0040700
0034700
0000400
0000040
,
27320000
8140000
0040000
0034100
0000382
0000373
,
3230000
1790000
001000
000100
000010
0000340
,
4000000
3510000
0034700
0040700
000010
0000340

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[27,8,0,0,0,0,32,14,0,0,0,0,0,0,40,34,0,0,0,0,0,1,0,0,0,0,0,0,38,37,0,0,0,0,2,3],[32,17,0,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,34,40,0,0,0,0,7,7,0,0,0,0,0,0,1,3,0,0,0,0,0,40] >;

D105M4(2) in GAP, Magma, Sage, TeX

D_{10}\rtimes_5M_4(2)
% in TeX

G:=Group("D10:5M4(2)");
// GroupNames label

G:=SmallGroup(320,463);
// by ID

G=gap.SmallGroup(320,463);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,701,120,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^5*b,d*b*d=a^3*b,d*c*d=c^5>;
// generators/relations

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