C5xC2^2.M4(2) - GroupNames

G = C₅×C₂².M₄(2) order 320 = 2⁶·5

Direct product of C₅ and C₂².M₄(2)

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₅×C₂².M₄(2), (C₂×C₄)⋊C₄₀, (C₂×C₂₀)⋊₇C₈, (C₂×C₂₀).₄₄₂D₄, C₂²⋊C₈.₁C₁₀, C₂².₃(C₂×C₄₀), (C₂²×C₄).₂C₂₀, (C₂²×C₂₀).₅C₄, C₂³.₂₂(C₂×C₂₀), C₁₀.₃₆(C₂²⋊C₈), C₁₀.₄₉(C₂³⋊C₄), (C₂×C₁₀).₃₃M₄(2), (C₂²×C₂₀).₂C₂², C₂².₃(C₅×M₄(2)), C₁₀.₁₅(C₄.₁₀D₄), (C₂×C₄⋊C₄).₁C₁₀, (C₂×C₄).₉₂(C₅×D₄), C₂.₄(C₅×C₂²⋊C₈), C₂.₂(C₅×C₂³⋊C₄), (C₁₀×C₄⋊C₄).₂₈C₂, (C₂×C₁₀).₄₉(C₂×C₈), (C₅×C₂²⋊C₈).₃C₂, (C₂²×C₄).₂(C₂×C₁₀), C₂.₁(C₅×C₄.₁₀D₄), C₂².₂₄(C₅×C₂²⋊C₄), (C₂²×C₁₀).₁₇₅(C₂×C₄), (C₂×C₁₀).₁₈₃(C₂²⋊C₄), SmallGroup(320,129)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×C₂².M₄(2)

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×C₂₀ — C₅×C₂²⋊C₈ — C₅×C₂².M₄(2)

Lower central

C₁ — C₂ — C₂² — C₅×C₂².M₄(2)

Upper central

C₁ — C₂×C₁₀ — C₂²×C₂₀ — C₅×C₂².M₄(2)

Generators and relations for C₅×C₂².M₄(2)
G = < a,b,c,d,e | a⁵=b²=c²=d⁸=1, e²=c, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=bcd⁵ >

Subgroups: 138 in 78 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, C₁₀, C₁₀, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂²⋊C₈, C₂×C₄⋊C₄, C₄₀, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂².M₄(2), C₅×C₄⋊C₄, C₂×C₄₀, C₂²×C₂₀, C₅×C₂²⋊C₈, C₁₀×C₄⋊C₄, C₅×C₂².M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₅, C₈, C₂×C₄, D₄, C₁₀, C₂²⋊C₄, C₂×C₈, M₄(2), C₂₀, C₂×C₁₀, C₂²⋊C₈, C₂³⋊C₄, C₄.₁₀D₄, C₄₀, C₂×C₂₀, C₅×D₄, C₂².M₄(2), C₅×C₂²⋊C₄, C₂×C₄₀, C₅×M₄(2), C₅×C₂²⋊C₈, C₅×C₂³⋊C₄, C₅×C₄.₁₀D₄, C₅×C₂².M₄(2)

Smallest permutation representation of C₅×C₂².M₄(2)
►On 160 points

Generators in S₁₆₀

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

(2 86)(4 88)(6 82)(8 84)(10 97)(12 99)(14 101)(16 103)(18 105)(20 107)(22 109)(24 111)(26 123)(28 125)(30 127)(32 121)(34 92)(36 94)(38 96)(40 90)(42 154)(44 156)(46 158)(48 160)(50 146)(52 148)(54 150)(56 152)(57 116)(59 118)(61 120)(63 114)(65 132)(67 134)(69 136)(71 130)(73 140)(75 142)(77 144)(79 138)

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

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

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

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

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

110 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	5C	5D	8A	···	8H	10A	···	10L	10M	···	10T	20A	···	20P	20Q	···	20AF	40A	···	40AF
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	5	5	5	5	8	···	8	10	···	10	10	···	10	20	···	20	20	···	20	40	···	40
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	1	1	1	1	4	···	4	1	···	1	2	···	2	2	···	2	4	···	4	4	···	4

110 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+								+				+	-
image	C₁	C₂	C₂	C₄	C₅	C₈	C₁₀	C₁₀	C₂₀	C₄₀	D₄	M₄(2)	C₅×D₄	C₅×M₄(2)	C₂³⋊C₄	C₄.₁₀D₄	C₅×C₂³⋊C₄	C₅×C₄.₁₀D₄
kernel	C₅×C₂².M₄(2)	C₅×C₂²⋊C₈	C₁₀×C₄⋊C₄	C₂²×C₂₀	C₂².M₄(2)	C₂×C₂₀	C₂²⋊C₈	C₂×C₄⋊C₄	C₂²×C₄	C₂×C₄	C₂×C₂₀	C₂×C₁₀	C₂×C₄	C₂²	C₁₀	C₁₀	C₂	C₂
# reps	1	2	1	4	4	8	8	4	16	32	2	2	8	8	1	1	4	4

Matrix representation of C₅×C₂².M₄(2) ►in GL₆(𝔽₄₁)

16	0	0	0	0	0
0	16	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	1

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	38	39	40	0
0	0	34	0	0	40

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

32	7	0	0	0	0
34	9	0	0	0	0
0	0	14	23	23	0
0	0	30	40	40	9
0	0	11	19	28	32
0	0	17	22	22	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	9	0	0	0
0	0	1	32	0	0
0	0	40	18	9	0
0	0	19	0	0	32

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,38,34,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,34,0,0,0,0,7,9,0,0,0,0,0,0,14,30,11,17,0,0,23,40,19,22,0,0,23,40,28,22,0,0,0,9,32,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,1,40,19,0,0,0,32,18,0,0,0,0,0,9,0,0,0,0,0,0,32] >;

C₅×C₂².M₄(2) in GAP, Magma, Sage, TeX

C_5\times C_2^2.M_4(2)

% in TeX

G:=Group("C5xC2^2.M4(2)");

// GroupNames label

G:=SmallGroup(320,129);

// by ID

G=gap.SmallGroup(320,129);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,2111,172]);

// Polycyclic

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

// generators/relations

G = C5×C22.M4(2) order 320 = 26·5

Direct product of C5 and C22.M4(2)

G = C₅×C₂².M₄(2) order 320 = 2⁶·5

Direct product of C₅ and C₂².M₄(2)