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G = M4(2)×Dic5order 320 = 26·5

Direct product of M4(2) and Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)×Dic5, C20.36C42, C4028(C2×C4), C86(C2×Dic5), C57(C4×M4(2)), C408C426C2, C4.6(C4×Dic5), (C8×Dic5)⋊29C2, (C2×C8).274D10, (C5×M4(2))⋊8C4, C4.Dic516C4, C23.52(C4×D5), C2.7(D5×M4(2)), C10.45(C2×C42), (C2×C10).28C42, (C4×Dic5).11C4, C22.6(C4×Dic5), (C2×C20).862C23, (C2×C40).232C22, C20.199(C22×C4), (C22×C4).344D10, (C2×M4(2)).18D5, (C10×M4(2)).5C2, C10.63(C2×M4(2)), C4.34(C22×Dic5), (C22×Dic5).20C4, (C22×C20).176C22, (C4×Dic5).314C22, C4.114(C2×C4×D5), C52C826(C2×C4), (C2×C4×Dic5).9C2, C2.13(C2×C4×Dic5), C22.62(C2×C4×D5), (C2×C4).156(C4×D5), (C2×C20).269(C2×C4), (C2×C4).46(C2×Dic5), (C2×C4).804(C22×D5), (C2×C4.Dic5).21C2, (C22×C10).130(C2×C4), (C2×C10).233(C22×C4), (C2×C52C8).329C22, (C2×Dic5).155(C2×C4), SmallGroup(320,744)

Series: Derived Chief Lower central Upper central

C1C10 — M4(2)×Dic5
C1C5C10C2×C10C2×C20C4×Dic5C2×C4×Dic5 — M4(2)×Dic5
C5C10 — M4(2)×Dic5
C1C2×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, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 334 in 142 conjugacy classes, 91 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×4], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2), C2×M4(2), C52C8 [×4], C40 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4×M4(2), C2×C52C8 [×2], C4.Dic5 [×4], C4×Dic5 [×2], C4×Dic5 [×2], C2×C40 [×2], C5×M4(2) [×4], C22×Dic5 [×2], C22×C20, C8×Dic5 [×2], C408C4 [×2], C2×C4.Dic5, C2×C4×Dic5, C10×M4(2), M4(2)×Dic5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], M4(2) [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C2×C42, C2×M4(2) [×2], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C4×M4(2), C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, D5×M4(2) [×2], C2×C4×Dic5, M4(2)×Dic5

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

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N4O4P4Q4R5A5B8A···8H8I···8P10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order1222224444444···44444558···88···810···101010101020···202020202040···40
size1111221111225···510101010222···210···102···244442···244444···4

80 irreducible representations

dim111111111122222224
type++++++++-+
imageC1C2C2C2C2C2C4C4C4C4D5M4(2)D10Dic5D10C4×D5C4×D5D5×M4(2)
kernelM4(2)×Dic5C8×Dic5C408C4C2×C4.Dic5C2×C4×Dic5C10×M4(2)C4.Dic5C4×Dic5C5×M4(2)C22×Dic5C2×M4(2)Dic5C2×C8M4(2)C22×C4C2×C4C23C2
# reps1221118484284821248

Matrix representation of M4(2)×Dic5 in GL4(𝔽41) generated by

40000
04000
001112
001830
,
40000
04000
004024
0001
,
04000
13500
0010
0001
,
392600
14200
00400
00040
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,11,18,0,0,12,30],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,24,1],[0,1,0,0,40,35,0,0,0,0,1,0,0,0,0,1],[39,14,0,0,26,2,0,0,0,0,40,0,0,0,0,40] >;

M4(2)×Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("M4(2)xDic5");
// GroupNames label

G:=SmallGroup(320,744);
// by ID

G=gap.SmallGroup(320,744);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,102,12550]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽