C10xC4:1D4 - GroupNames

G = C₁₀×C₄⋊₁D₄ order 320 = 2⁶·5

Direct product of C₁₀ and C₄⋊₁D₄

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

Aliases: C₁₀×C₄⋊₁D₄, C₄⋊₁(D₄×C₁₀), (C₂×C₂₀)⋊₃₃D₄, C₂₀⋊₁₂(C₂×D₄), (C₂×C₄²)⋊₁₁C₁₀, C₄²⋊₁₈(C₂×C₁₀), (C₄×C₂₀)⋊₅₉C₂², (C₂²×D₄)⋊₅C₁₀, (D₄×C₁₀)⋊₆₃C₂², C₂⁴.₁₅(C₂×C₁₀), C₂².₆₃(D₄×C₁₀), (C₂×C₂₀).₉₆₁C₂³, (C₂×C₁₀).₃₅₀C₂⁴, C₁₀.₁₈₆(C₂²×D₄), C₂³.₈(C₂²×C₁₀), (C₂³×C₁₀).₁₅C₂², C₂².₂₄(C₂³×C₁₀), (C₂²×C₁₀).₈₈C₂³, (C₂²×C₂₀).₅₉₆C₂², (C₂×C₄×C₂₀)⋊₂₄C₂, (C₂×C₄)⋊₇(C₅×D₄), (D₄×C₂×C₁₀)⋊₂₀C₂, C₂.₁₀(D₄×C₂×C₁₀), (C₂×D₄)⋊₁₁(C₂×C₁₀), (C₂×C₁₀).₆₈₄(C₂×D₄), (C₂×C₄).₁₃₆(C₂²×C₁₀), (C₂²×C₄).₁₂₄(C₂×C₁₀), SmallGroup(320,1532)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₁₀×C₄⋊₁D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₅×C₄⋊₁D₄ — C₁₀×C₄⋊₁D₄

Lower central

C₁ — C₂² — C₁₀×C₄⋊₁D₄

Upper central

C₁ — C₂²×C₁₀ — C₁₀×C₄⋊₁D₄

Generators and relations for C₁₀×C₄⋊₁D₄
G = < a,b,c,d | a¹⁰=b⁴=c⁴=d²=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b^-1, dcd=c^-1 >

Subgroups: 882 in 498 conjugacy classes, 210 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₄², C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₄², C₄⋊₁D₄, C₂²×D₄, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₄⋊₁D₄, C₄×C₂₀, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₂×C₄×C₂₀, C₅×C₄⋊₁D₄, D₄×C₂×C₁₀, C₁₀×C₄⋊₁D₄
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂⁴, C₂×C₁₀, C₄⋊₁D₄, C₂²×D₄, C₅×D₄, C₂²×C₁₀, C₂×C₄⋊₁D₄, D₄×C₁₀, C₂³×C₁₀, C₅×C₄⋊₁D₄, D₄×C₂×C₁₀, C₁₀×C₄⋊₁D₄

Smallest permutation representation of C₁₀×C₄⋊₁D₄
►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 16 62 154)(2 17 63 155)(3 18 64 156)(4 19 65 157)(5 20 66 158)(6 11 67 159)(7 12 68 160)(8 13 69 151)(9 14 70 152)(10 15 61 153)(21 41 40 55)(22 42 31 56)(23 43 32 57)(24 44 33 58)(25 45 34 59)(26 46 35 60)(27 47 36 51)(28 48 37 52)(29 49 38 53)(30 50 39 54)(71 139 99 111)(72 140 100 112)(73 131 91 113)(74 132 92 114)(75 133 93 115)(76 134 94 116)(77 135 95 117)(78 136 96 118)(79 137 97 119)(80 138 98 120)(81 145 105 121)(82 146 106 122)(83 147 107 123)(84 148 108 124)(85 149 109 125)(86 150 110 126)(87 141 101 127)(88 142 102 128)(89 143 103 129)(90 144 104 130)

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

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

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

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

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

140 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4L	5A	5B	5C	5D	10A	···	10AB	10AC	···	10BH	20A	···	20AV
order	1	2	···	2	2	···	2	4	···	4	5	5	5	5	10	···	10	10	···	10	20	···	20
size	1	1	···	1	4	···	4	2	···	2	1	1	1	1	1	···	1	4	···	4	2	···	2

140 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	C₅×D₄
kernel	C₁₀×C₄⋊₁D₄	C₂×C₄×C₂₀	C₅×C₄⋊₁D₄	D₄×C₂×C₁₀	C₂×C₄⋊₁D₄	C₂×C₄²	C₄⋊₁D₄	C₂²×D₄	C₂×C₂₀	C₂×C₄
# reps	1	1	8	6	4	4	32	24	12	48

Matrix representation of C₁₀×C₄⋊₁D₄ ►in GL₅(𝔽₄₁)

40	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	37	0
0	0	0	0	37

40	0	0	0	0
0	40	2	0	0
0	40	1	0	0
0	0	0	34	2
0	0	0	16	7

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	34	2
0	0	0	16	7

40	0	0	0	0
0	1	39	0	0
0	0	40	0	0
0	0	0	40	0
0	0	0	34	1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,37,0,0,0,0,0,37],[40,0,0,0,0,0,40,40,0,0,0,2,1,0,0,0,0,0,34,16,0,0,0,2,7],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,16,0,0,0,2,7],[40,0,0,0,0,0,1,0,0,0,0,39,40,0,0,0,0,0,40,34,0,0,0,0,1] >;

C₁₀×C₄⋊₁D₄ in GAP, Magma, Sage, TeX

C_{10}\times C_4\rtimes_1D_4

% in TeX

G:=Group("C10xC4:1D4");

// GroupNames label

G:=SmallGroup(320,1532);

// by ID

G=gap.SmallGroup(320,1532);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,856]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=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 = C10×C4⋊1D4 order 320 = 26·5

Direct product of C10 and C4⋊1D4

G = C₁₀×C₄⋊₁D₄ order 320 = 2⁶·5

Direct product of C₁₀ and C₄⋊₁D₄