D10:2M4(2) - GroupNames

G = D₁₀⋊₂M₄(2) order 320 = 2⁶·5

2^nd semidirect product of D₁₀ and M₄(2) acting via M₄(2)/C₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀⋊₂M₄(2), C₅⋊C₈⋊₆D₄, C₄⋊C₄.₅F₅, C₅⋊₃(C₈⋊₉D₄), C₁₀.₉(C₄×D₄), C₂.₁₁(D₄×F₅), (C₂×D₂₀).₈C₄, D₁₀⋊C₈⋊₉C₂, C₂.₄(Q₈.F₅), Dic₅⋊C₈⋊₂C₂, D₁₀⋊C₄.₉C₄, C₁₀.₂₀(C₈○D₄), Dic₅.₇₀(C₂×D₄), D₂₀⋊₈C₄.₁₇C₂, C₁₀.C₄²⋊₃C₂, C₁₀.₁₃(C₂×M₄(2)), Dic₅.₅₅(C₄○D₄), C₂².₇₆(C₂²×F₅), C₂.₁₃(D₅⋊M₄(2)), (C₄×Dic₅).₆₈C₂², (C₂×Dic₅).₃₃₁C₂³, (C₅×C₄⋊C₄).₈C₄, (C₂×D₅⋊C₈)⋊₁₁C₂, (C₂×C₄.F₅)⋊₁₁C₂, (C₂×C₄).₂₅(C₂×F₅), (C₂×C₂₀).₂₂(C₂×C₄), (C₂×C₅⋊C₈).₂₈C₂², (C₂×C₄×D₅).₂₉₀C₂², (C₂×C₁₀).₄₂(C₂²×C₄), (C₂×Dic₅).₅₇(C₂×C₄), (C₂²×D₅).₄₈(C₂×C₄), SmallGroup(320,1042)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — D₁₀⋊₂M₄(2)

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₂×Dic₅ — C₂×C₅⋊C₈ — C₂×D₅⋊C₈ — D₁₀⋊₂M₄(2)

Lower central

C₅ — C₂×C₁₀ — D₁₀⋊₂M₄(2)

Upper central

C₁ — C₂² — C₄⋊C₄

Generators and relations for D₁₀⋊₂M₄(2)
G = < a,b,c,d | a¹⁰=b²=c⁸=d²=1, bab=dad=a^-1, cac^-1=a⁷, cbc^-1=ab, dbd=a³b, dcd=c⁵ >

Subgroups: 474 in 124 conjugacy classes, 46 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, D₅, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, Dic₅, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₄×D₄, C₂²×C₈, C₂×M₄(2), C₅⋊C₈, C₅⋊C₈, C₄×D₅, D₂₀, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₈⋊₉D₄, C₄×Dic₅, D₁₀⋊C₄, C₅×C₄⋊C₄, D₅⋊C₈, C₄.F₅, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂×D₂₀, C₁₀.C₄², D₁₀⋊C₈, Dic₅⋊C₈, D₂₀⋊₈C₄, C₂×D₅⋊C₈, C₂×C₄.F₅, D₁₀⋊₂M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂×M₄(2), C₈○D₄, C₂×F₅, C₈⋊₉D₄, C₂²×F₅, D₅⋊M₄(2), D₄×F₅, Q₈.F₅, D₁₀⋊₂M₄(2)

Smallest permutation representation of D₁₀⋊₂M₄(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 73)(2 72)(3 71)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 107)(12 106)(13 105)(14 104)(15 103)(16 102)(17 101)(18 110)(19 109)(20 108)(21 85)(22 84)(23 83)(24 82)(25 81)(26 90)(27 89)(28 88)(29 87)(30 86)(31 63)(32 62)(33 61)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 60)(50 59)(91 158)(92 157)(93 156)(94 155)(95 154)(96 153)(97 152)(98 151)(99 160)(100 159)(111 145)(112 144)(113 143)(114 142)(115 141)(116 150)(117 149)(118 148)(119 147)(120 146)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 140)(128 139)(129 138)(130 137)

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

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	5	8A	···	8H	8I	8J	8K	8L	10A	10B	10C	20A	···	20F
order	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	5	8	···	8	8	8	8	8	10	10	10	20	···	20
size	1	1	1	1	10	10	20	2	2	4	4	5	5	5	5	20	4	10	···	10	20	20	20	20	4	4	4	8	···	8

38 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	8	8
type	+	+	+	+	+	+	+				+				+	+		+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	C₄○D₄	M₄(2)	C₈○D₄	F₅	C₂×F₅	D₅⋊M₄(2)	D₄×F₅	Q₈.F₅
kernel	D₁₀⋊₂M₄(2)	C₁₀.C₄²	D₁₀⋊C₈	Dic₅⋊C₈	D₂₀⋊₈C₄	C₂×D₅⋊C₈	C₂×C₄.F₅	D₁₀⋊C₄	C₅×C₄⋊C₄	C₂×D₂₀	C₅⋊C₈	Dic₅	D₁₀	C₁₀	C₄⋊C₄	C₂×C₄	C₂	C₂	C₂
# reps	1	1	2	1	1	1	1	4	2	2	2	2	4	4	1	3	4	1	1

Matrix representation of D₁₀⋊₂M₄(2) ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	40	1
0	0	0	0	40	0
0	0	1	0	40	0
0	0	0	1	40	0

0	40	0	0	0	0
40	0	0	0	0	0
0	0	38	22	19	3
0	0	19	0	22	3
0	0	38	3	22	0
0	0	0	3	19	22

40	0	0	0	0	0
0	1	0	0	0	0
0	0	1	9	11	11
0	0	12	18	12	20
0	0	21	29	23	29
0	0	30	30	32	40

1	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	0	1	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,38,19,38,0,0,0,22,0,3,3,0,0,19,22,22,19,0,0,3,3,0,22],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,21,30,0,0,9,18,29,30,0,0,11,12,23,32,0,0,11,20,29,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;

D₁₀⋊₂M₄(2) in GAP, Magma, Sage, TeX

D_{10}\rtimes_2M_4(2)

% in TeX

G:=Group("D10:2M4(2)");

// GroupNames label

G:=SmallGroup(320,1042);

// by ID

G=gap.SmallGroup(320,1042);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,136,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a*b,d*b*d=a^3*b,d*c*d=c^5>;

// generators/relations

G = D10⋊2M4(2) order 320 = 26·5

2nd semidirect product of D10 and M4(2) acting via M4(2)/C4=C22

G = D₁₀⋊₂M₄(2) order 320 = 2⁶·5

2^nd semidirect product of D₁₀ and M₄(2) acting via M₄(2)/C₄=C₂²