C4^2.15F5 - GroupNames

G = C₄².₁₅F₅ order 320 = 2⁶·5

12^nd non-split extension by C₄² of F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₁₅F₅, D₁₀.₉M₄(2), C₂₀.₁₁M₄(2), (C₄×C₂₀).₇C₄, C₂₀⋊C₈⋊₈C₂, C₄.₈(C₄.F₅), D₁₀⋊C₈.₄C₂, C₅⋊₁(C₄².₆C₄), (C₄×Dic₅).₂₇C₄, (D₅×C₄²).₂₀C₂, C₁₀.₈(C₂×M₄(2)), C₁₀.C₄²⋊₅C₂, C₂.₉(D₅⋊M₄(2)), C₁₀.₃(C₄²⋊C₂), Dic₅.₂₅(C₄○D₄), C₂².₆₄(C₂²×F₅), (C₄×Dic₅).₃₂₂C₂², (C₂×Dic₅).₃₁₉C₂³, C₂.₈(D₁₀.C₂³), (C₂×C₄×D₅).₃₀C₄, C₂.₇(C₂×C₄.F₅), (C₂×C₅⋊C₈).₃C₂², (C₂×C₄).₉₉(C₂×F₅), (C₂×C₂₀).₁₂₃(C₂×C₄), (C₂×C₄×D₅).₃₅₈C₂², (C₂×C₁₀).₂₁(C₂²×C₄), (C₂×Dic₅).₁₆₉(C₂×C₄), (C₂²×D₅).₁₂₁(C₂×C₄), SmallGroup(320,1021)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₄².₁₅F₅

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₂×Dic₅ — C₂×C₅⋊C₈ — D₁₀⋊C₈ — C₄².₁₅F₅

Lower central

C₅ — C₂×C₁₀ — C₄².₁₅F₅

Upper central

C₁ — C₂² — C₄²

Generators and relations for C₄².₁₅F₅
G = < a,b,c,d | a⁴=b⁴=c⁵=1, d⁴=a², ab=ba, ac=ca, dad^-1=ab², bc=cb, dbd^-1=a²b, dcd^-1=c³ >

Subgroups: 378 in 110 conjugacy classes, 48 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₄², C₄², C₂×C₈, C₂²×C₄, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₄².₆C₄, C₄×Dic₅, C₄×C₂₀, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂₀⋊C₈, C₁₀.C₄², D₁₀⋊C₈, D₅×C₄², C₄².₁₅F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, M₄(2), C₂²×C₄, C₄○D₄, F₅, C₄²⋊C₂, C₂×M₄(2), C₂×F₅, C₄².₆C₄, C₄.F₅, C₂²×F₅, C₂×C₄.F₅, D₅⋊M₄(2), D₁₀.C₂³, C₄².₁₅F₅

Smallest permutation representation of C₄².₁₅F₅
►On 160 points

Generators in S₁₆₀

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

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

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

(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)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	4	4	4	4	4
type	+	+	+	+	+							+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄○D₄	M₄(2)	M₄(2)	F₅	C₂×F₅	C₄.F₅	D₅⋊M₄(2)	D₁₀.C₂³
kernel	C₄².₁₅F₅	C₂₀⋊C₈	C₁₀.C₄²	D₁₀⋊C₈	D₅×C₄²	C₄×Dic₅	C₄×C₂₀	C₂×C₄×D₅	Dic₅	C₂₀	D₁₀	C₄²	C₂×C₄	C₄	C₂	C₂
# reps	1	2	2	2	1	2	2	4	4	4	4	1	3	4	4	4

Matrix representation of C₄².₁₅F₅ ►in GL₆(𝔽₄₁)

32	0	0	0	0	0
0	32	0	0	0	0
0	0	22	0	3	3
0	0	38	19	38	0
0	0	0	38	19	38
0	0	3	3	0	22

0	40	0	0	0	0
40	0	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

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

6	39	0	0	0	0
2	35	0	0	0	0
0	0	7	35	24	13
0	0	30	19	6	13
0	0	28	35	22	11
0	0	28	17	6	34

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[6,2,0,0,0,0,39,35,0,0,0,0,0,0,7,30,28,28,0,0,35,19,35,17,0,0,24,6,22,6,0,0,13,13,11,34] >;

C₄².₁₅F₅ in GAP, Magma, Sage, TeX

C_4^2._{15}F_5

% in TeX

G:=Group("C4^2.15F5");

// GroupNames label

G:=SmallGroup(320,1021);

// by ID

G=gap.SmallGroup(320,1021);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,232,758,100,136,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;

// generators/relations

G = C42.15F5 order 320 = 26·5

12nd non-split extension by C42 of F5 acting via F5/D5=C2

G = C₄².₁₅F₅ order 320 = 2⁶·5

12^nd non-split extension by C₄² of F₅ acting via F₅/D₅=C₂