C5xC4:Q8 - GroupNames

G = C₅×C₄⋊Q₈ order 160 = 2⁵·5

Direct product of C₅ and C₄⋊Q₈

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

Aliases: C₅×C₄⋊Q₈, C₂₀⋊₄Q₈, C₂₀.₄₀D₄, C₄².₅C₁₀, C₄⋊(C₅×Q₈), C₄.₅(C₅×D₄), C₄⋊C₄.₅C₁₀, C₂.₅(Q₈×C₁₀), (C₄×C₂₀).₁₁C₂, C₁₀.₇₃(C₂×D₄), C₂.₁₀(D₄×C₁₀), (C₂×Q₈).₃C₁₀, (Q₈×C₁₀).₈C₂, C₁₀.₂₂(C₂×Q₈), (C₂×C₁₀).₈₃C₂³, (C₂×C₂₀).₁₂₆C₂², C₂².₁₈(C₂²×C₁₀), (C₅×C₄⋊C₄).₁₂C₂, (C₂×C₄).₉(C₂×C₁₀), SmallGroup(160,189)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×C₄⋊Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₁₀ — C₂×C₂₀ — Q₈×C₁₀ — C₅×C₄⋊Q₈

Lower central

C₁ — C₂² — C₅×C₄⋊Q₈

Upper central

C₁ — C₂×C₁₀ — C₅×C₄⋊Q₈

Generators and relations for C₅×C₄⋊Q₈
G = < a,b,c,d | a⁵=b⁴=c⁴=1, d²=c², ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1, dcd^-1=c^-1 >

Subgroups: 84 in 68 conjugacy classes, 52 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₂×C₄, C₂×C₄, Q₈, C₁₀, C₁₀, C₄², C₄⋊C₄, C₂×Q₈, C₂₀, C₂₀, C₂×C₁₀, C₄⋊Q₈, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, C₄×C₂₀, C₅×C₄⋊C₄, Q₈×C₁₀, C₅×C₄⋊Q₈
Quotients: C₁, C₂, C₂², C₅, D₄, Q₈, C₂³, C₁₀, C₂×D₄, C₂×Q₈, C₂×C₁₀, C₄⋊Q₈, C₅×D₄, C₅×Q₈, C₂²×C₁₀, D₄×C₁₀, Q₈×C₁₀, C₅×C₄⋊Q₈

Smallest permutation representation of C₅×C₄⋊Q₈
►Regular action 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 95 27 81)(2 91 28 82)(3 92 29 83)(4 93 30 84)(5 94 26 85)(6 100 14 106)(7 96 15 107)(8 97 11 108)(9 98 12 109)(10 99 13 110)(16 101 156 115)(17 102 157 111)(18 103 158 112)(19 104 159 113)(20 105 160 114)(21 87 35 76)(22 88 31 77)(23 89 32 78)(24 90 33 79)(25 86 34 80)(36 67 47 56)(37 68 48 57)(38 69 49 58)(39 70 50 59)(40 66 46 60)(41 75 55 61)(42 71 51 62)(43 72 52 63)(44 73 53 64)(45 74 54 65)(116 136 127 147)(117 137 128 148)(118 138 129 149)(119 139 130 150)(120 140 126 146)(121 141 135 155)(122 142 131 151)(123 143 132 152)(124 144 133 153)(125 145 134 154)

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

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

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

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

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

C₅×C₄⋊Q₈ is a maximal subgroup of
C₂₀.₅Q₁₆ C₂₀.₁₀D₈ C₄².₃Dic₅ C₂₀.₁₇D₈ C₂₀.SD₁₆ C₄².₇₆D₁₀ C₂₀.Q₁₆ C₄².₇₇D₁₀ C₂₀⋊₅SD₁₆ D₂₀⋊₅Q₈ C₂₀⋊₆SD₁₆ C₄².₈₀D₁₀ D₂₀⋊₆Q₈ C₂₀.D₈ C₄².₈₂D₁₀ C₂₀⋊Q₁₆ Dic₁₀⋊₅Q₈ C₂₀⋊₃Q₁₆ C₂₀.₁₁Q₁₆ Dic₁₀⋊₆Q₈ D₂₀.₁₅D₄ Dic₁₀⋊₈Q₈ Dic₁₀⋊₉Q₈ C₄².₁₇₁D₁₀ C₄².₂₄₀D₁₀ D₂₀⋊₁₂D₄ D₂₀⋊₈Q₈ C₄².₂₄₁D₁₀ C₄².₁₇₄D₁₀ D₂₀⋊₉Q₈ C₄².₁₇₆D₁₀ C₄².₁₇₇D₁₀ C₄².₁₇₈D₁₀ C₄².₁₇₉D₁₀ C₄².₁₈₀D₁₀ C₅×D₄×Q₈ C₅×Q₈²

70 conjugacy classes

class	1	2A	2B	2C	4A	···	4F	4G	4H	4I	4J	5A	5B	5C	5D	10A	···	10L	20A	···	20X	20Y	···	20AN
order	1	2	2	2	4	···	4	4	4	4	4	5	5	5	5	10	···	10	20	···	20	20	···	20
size	1	1	1	1	2	···	2	4	4	4	4	1	1	1	1	1	···	1	2	···	2	4	···	4

70 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+					+	-
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	Q₈	C₅×D₄	C₅×Q₈
kernel	C₅×C₄⋊Q₈	C₄×C₂₀	C₅×C₄⋊C₄	Q₈×C₁₀	C₄⋊Q₈	C₄²	C₄⋊C₄	C₂×Q₈	C₂₀	C₂₀	C₄	C₄
# reps	1	1	4	2	4	4	16	8	2	4	8	16

Matrix representation of C₅×C₄⋊Q₈ ►in GL₅(𝔽₄₁)

37	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

40	0	0	0	0
0	0	1	0	0
0	40	0	0	0
0	0	0	40	2
0	0	0	40	1

40	0	0	0	0
0	6	2	0	0
0	2	35	0	0
0	0	0	33	12
0	0	0	39	8

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,40,0,0,0,2,1],[40,0,0,0,0,0,6,2,0,0,0,2,35,0,0,0,0,0,33,39,0,0,0,12,8] >;

C₅×C₄⋊Q₈ in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes Q_8

% in TeX

G:=Group("C5xC4:Q8");

// GroupNames label

G:=SmallGroup(160,189);

// by ID

G=gap.SmallGroup(160,189);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,505,247,1514,374]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;

// generators/relations

G = C5×C4⋊Q8 order 160 = 25·5

Direct product of C5 and C4⋊Q8

G = C₅×C₄⋊Q₈ order 160 = 2⁵·5

Direct product of C₅ and C₄⋊Q₈