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G = M4(2)⋊Dic5order 320 = 26·5

2nd semidirect product of M4(2) and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊2Dic5, (C2×C20).9Q8, C20.40(C4⋊C4), (C2×C4).128D20, (C2×C20).468D4, C4.Dic511C4, C4.3(C4⋊Dic5), (C2×C4).4Dic10, C23.43(C4×D5), (C5×M4(2))⋊11C4, (C2×C10).25C42, (C22×C4).61D10, (C2×M4(2)).5D5, C22.3(C4×Dic5), C54(C22.C42), (C10×M4(2)).9C2, C4.17(C23.D5), C4.34(D10⋊C4), (C22×Dic5).3C4, C20.133(C22⋊C4), C4.24(C10.D4), C10.12(C4.D4), C2.2(C20.46D4), C2.2(C4.12D20), C10.10(C4.10D4), (C22×C20).123C22, C22.5(C10.D4), C22.41(D10⋊C4), C10.32(C2.C42), C2.13(C10.10C42), (C2×C4).20(C4×D5), (C2×C10).34(C4⋊C4), (C2×C20).234(C2×C4), (C2×C4).21(C5⋊D4), (C2×C4⋊Dic5).29C2, (C2×C4).14(C2×Dic5), (C22×C10).99(C2×C4), (C2×C4.Dic5).10C2, (C2×C10).117(C22⋊C4), SmallGroup(320,112)

Series: Derived Chief Lower central Upper central

C1C2×C10 — M4(2)⋊Dic5
C1C5C10C2×C10C2×C20C22×C20C2×C4⋊Dic5 — M4(2)⋊Dic5
C5C10C2×C10 — M4(2)⋊Dic5
C1C22C22×C4C2×M4(2)

Generators and relations for M4(2)⋊Dic5
 G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 310 in 98 conjugacy classes, 51 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×6], C2×C4 [×4], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], Dic5 [×2], C20 [×4], C2×C10 [×3], C2×C10 [×2], C2×C4⋊C4, C2×M4(2), C2×M4(2), C52C8 [×2], C40 [×2], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C22.C42, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4⋊Dic5 [×2], C2×C40, C5×M4(2) [×2], C5×M4(2), C22×Dic5 [×2], C22×C20, C2×C4.Dic5, C2×C4⋊Dic5, C10×M4(2), M4(2)⋊Dic5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, C4.D4, C4.10D4, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C22.C42, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C20.46D4, C4.12D20, C10.10C42, M4(2)⋊Dic5

Smallest permutation representation of M4(2)⋊Dic5
On 160 points
Generators in S160
(1 79 34 63 28 90 48 59)(2 80 35 64 29 81 49 60)(3 71 36 65 30 82 50 51)(4 72 37 66 21 83 41 52)(5 73 38 67 22 84 42 53)(6 74 39 68 23 85 43 54)(7 75 40 69 24 86 44 55)(8 76 31 70 25 87 45 56)(9 77 32 61 26 88 46 57)(10 78 33 62 27 89 47 58)(11 122 142 102 151 111 131 91)(12 123 143 103 152 112 132 92)(13 124 144 104 153 113 133 93)(14 125 145 105 154 114 134 94)(15 126 146 106 155 115 135 95)(16 127 147 107 156 116 136 96)(17 128 148 108 157 117 137 97)(18 129 149 109 158 118 138 98)(19 130 150 110 159 119 139 99)(20 121 141 101 160 120 140 100)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 156)(12 157)(13 158)(14 159)(15 160)(16 151)(17 152)(18 153)(19 154)(20 155)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(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 91 6 96)(2 100 7 95)(3 99 8 94)(4 98 9 93)(5 97 10 92)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444558888888810···101010101020···202020202040···40
size111122222220202020224444202020202···244442···244444···4

62 irreducible representations

dim111111122222222224444
type+++++-+-+-++-+-
imageC1C2C2C2C4C4C4D4Q8D5Dic5D10Dic10C4×D5D20C5⋊D4C4×D5C4.D4C4.10D4C20.46D4C4.12D20
kernelM4(2)⋊Dic5C2×C4.Dic5C2×C4⋊Dic5C10×M4(2)C4.Dic5C5×M4(2)C22×Dic5C2×C20C2×C20C2×M4(2)M4(2)C22×C4C2×C4C2×C4C2×C4C2×C4C23C10C10C2C2
# reps111144431242444841144

Matrix representation of M4(2)⋊Dic5 in GL6(𝔽41)

900000
090000
000010
000001
00113200
0093000
,
100000
010000
001000
000100
0000400
0000040
,
2500000
2230000
0034100
0040000
0000341
0000400
,
6350000
13350000
0022100
00353900
0000221
00003539

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,11,9,0,0,0,0,32,30,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[25,2,0,0,0,0,0,23,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,34,40,0,0,0,0,1,0],[6,13,0,0,0,0,35,35,0,0,0,0,0,0,2,35,0,0,0,0,21,39,0,0,0,0,0,0,2,35,0,0,0,0,21,39] >;

M4(2)⋊Dic5 in GAP, Magma, Sage, TeX

M_4(2)\rtimes {\rm Dic}_5
% in TeX

G:=Group("M4(2):Dic5");
// GroupNames label

G:=SmallGroup(320,112);
// by ID

G=gap.SmallGroup(320,112);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,136,851,12550]);
// Polycyclic

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

׿
×
𝔽