C2xC20:4D4 - GroupNames

G = C₂×C₂₀⋊₄D₄ order 320 = 2⁶·5

Direct product of C₂ and C₂₀⋊₄D₄

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

Aliases: C₂×C₂₀⋊₄D₄, C₄²⋊₃₉D₁₀, C₄⋊₂(C₂×D₂₀), (C₂×C₄)⋊₇D₂₀, (C₂×C₂₀)⋊₃₀D₄, C₂₀⋊₁₀(C₂×D₄), (C₂×C₄²)⋊₉D₅, C₁₀⋊₁(C₄⋊₁D₄), (C₄×C₂₀)⋊₅₀C₂², (C₂²×D₂₀)⋊₄C₂, C₁₀.₃(C₂²×D₄), C₂.₅(C₂²×D₂₀), (C₂×D₂₀)⋊₄₂C₂², (C₂×C₁₀).₁₉C₂⁴, C₂².₆₄(C₂×D₂₀), (C₂×C₂₀).₇₈₀C₂³, (C₂²×C₄).₄₃₈D₁₀, (C₂²×D₅).₁C₂³, C₂².₆₂(C₂³×D₅), (C₂³×D₅).₂₇C₂², C₂³.₃₁₅(C₂²×D₅), (C₂²×C₂₀).₅₂₃C₂², (C₂²×C₁₀).₃₈₁C₂³, (C₂×C₄×C₂₀)⋊₈C₂, C₅⋊₁(C₂×C₄⋊₁D₄), (C₂×C₁₀).₁₇₀(C₂×D₄), (C₂×C₄).₇₂₉(C₂²×D₅), SmallGroup(320,1147)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×C₂₀⋊₄D₄

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂³×D₅ — C₂²×D₂₀ — C₂×C₂₀⋊₄D₄

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀⋊₄D₄

Upper central

C₁ — C₂³ — C₂×C₄²

Generators and relations for C₂×C₂₀⋊₄D₄
G = < a,b,c,d | a²=b⁴=c²⁰=d²=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b^-1, dcd=c^-1 >

Subgroups: 2430 in 498 conjugacy classes, 159 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₄², C₂²×C₄, C₂×D₄, C₂⁴, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₄², C₄⋊₁D₄, C₂²×D₄, D₂₀, C₂×C₂₀, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂×C₄⋊₁D₄, C₄×C₂₀, C₂×D₂₀, C₂×D₂₀, C₂²×C₂₀, C₂³×D₅, C₂₀⋊₄D₄, C₂×C₄×C₂₀, C₂²×D₂₀, C₂×C₂₀⋊₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₄⋊₁D₄, C₂²×D₄, D₂₀, C₂²×D₅, C₂×C₄⋊₁D₄, C₂×D₂₀, C₂³×D₅, C₂₀⋊₄D₄, C₂²×D₂₀, C₂×C₂₀⋊₄D₄

Smallest permutation representation of C₂×C₂₀⋊₄D₄
►On 160 points

Generators in S₁₆₀

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

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

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

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

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

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

92 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4L	5A	5B	10A	···	10N	20A	···	20AV
order	1	2	···	2	2	···	2	4	···	4	5	5	10	···	10	20	···	20
size	1	1	···	1	20	···	20	2	···	2	2	2	2	···	2	2	···	2

92 irreducible representations

dim	1	1	1	1	2	2	2	2	2
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	D₄	D₅	D₁₀	D₁₀	D₂₀
kernel	C₂×C₂₀⋊₄D₄	C₂₀⋊₄D₄	C₂×C₄×C₂₀	C₂²×D₂₀	C₂×C₂₀	C₂×C₄²	C₄²	C₂²×C₄	C₂×C₄
# reps	1	8	1	6	12	2	8	6	48

Matrix representation of C₂×C₂₀⋊₄D₄ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	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

2	32	0	0	0	0
37	39	0	0	0	0
0	0	2	28	0	0
0	0	13	39	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	1	0	0	0	0
5	6	0	0	0	0
0	0	0	40	0	0
0	0	1	6	0	0
0	0	0	0	25	30
0	0	0	0	27	39

28	14	0	0	0	0
29	13	0	0	0	0
0	0	16	39	0	0
0	0	25	25	0	0
0	0	0	0	28	27
0	0	0	0	12	13

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[2,37,0,0,0,0,32,39,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,5,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,25,27,0,0,0,0,30,39],[28,29,0,0,0,0,14,13,0,0,0,0,0,0,16,25,0,0,0,0,39,25,0,0,0,0,0,0,28,12,0,0,0,0,27,13] >;

C₂×C₂₀⋊₄D₄ in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_4D_4

% in TeX

G:=Group("C2xC20:4D4");

// GroupNames label

G:=SmallGroup(320,1147);

// by ID

G=gap.SmallGroup(320,1147);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,80,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C2×C20⋊4D4 order 320 = 26·5

Direct product of C2 and C20⋊4D4

G = C₂×C₂₀⋊₄D₄ order 320 = 2⁶·5

Direct product of C₂ and C₂₀⋊₄D₄