C5:C16.C2^2 - GroupNames

G = C₅⋊C₁₆.C₂² order 320 = 2⁶·5

The non-split extension by C₅⋊C₁₆ of C₂² acting via C₂²/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅⋊C₁₆.C₂², C₅⋊₂(D₄○C₁₆), D₄.₂(C₅⋊C₈), (C₅×D₄).₂C₈, C₄○D₄.₃F₅, Q₈.₂(C₅⋊C₈), (C₅×Q₈).₂C₈, C₂₀.₁₀(C₂×C₈), C₂₀.C₈⋊₄C₂, C₄.Dic₅.₄C₄, C₄.₅₆(C₂²×F₅), C₂₀.₉₆(C₂²×C₄), C₁₀.₂₃(C₂²×C₈), C₅⋊₂C₈.₄₁C₂³, D₄.Dic₅.₃C₂, C₄.₅(C₂×C₅⋊C₈), (C₂×C₅⋊C₁₆)⋊₄C₂, C₂².₁(C₂×C₅⋊C₈), C₂.₈(C₂²×C₅⋊C₈), (C₂×C₁₀).₂(C₂×C₈), (C₅×C₄○D₄).₂C₄, (C₂×C₄).₈₈(C₂×F₅), (C₂×C₂₀).₆₅(C₂×C₄), C₅⋊₂C₈.₂₂(C₂×C₄), (C₂×C₅⋊₂C₈).₁₈₉C₂², SmallGroup(320,1129)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₅⋊C₁₆.C₂²

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₅⋊₂C₈ — C₅⋊C₁₆ — C₂×C₅⋊C₁₆ — C₅⋊C₁₆.C₂²

Lower central

C₅ — C₁₀ — C₅⋊C₁₆.C₂²

Upper central

C₁ — C₄ — C₄○D₄

Generators and relations for C₅⋊C₁₆.C₂²
G = < a,b,c,d | a⁵=b¹⁶=c²=1, d²=b², bab^-1=dad^-1=a³, ac=ca, cbc=b⁹, bd=db, cd=dc >

Subgroups: 178 in 84 conjugacy classes, 56 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, C₂×C₄, D₄, Q₈, C₁₀, C₁₀, C₁₆, C₂×C₈, M₄(2), C₄○D₄, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₆, M₅(2), C₈○D₄, C₅⋊₂C₈, C₅⋊₂C₈, C₂×C₂₀, C₅×D₄, C₅×Q₈, D₄○C₁₆, C₅⋊C₁₆, C₅⋊C₁₆, C₂×C₅⋊₂C₈, C₄.Dic₅, C₅×C₄○D₄, C₂×C₅⋊C₁₆, C₂₀.C₈, D₄.Dic₅, C₅⋊C₁₆.C₂²
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, C₂×C₈, C₂²×C₄, F₅, C₂²×C₈, C₅⋊C₈, C₂×F₅, D₄○C₁₆, C₂×C₅⋊C₈, C₂²×F₅, C₂²×C₅⋊C₈, C₅⋊C₁₆.C₂²

Smallest permutation representation of C₅⋊C₁₆.C₂²
►On 160 points

Generators in S₁₆₀

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

(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)

(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(66 74)(68 76)(70 78)(72 80)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)(113 121)(115 123)(117 125)(119 127)(130 138)(132 140)(134 142)(136 144)(145 153)(147 155)(149 157)(151 159)

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	5	8A	8B	8C	8D	8E	···	8J	10A	10B	10C	10D	16A	···	16H	16I	···	16T	20A	20B	20C	20D	20E
order	1	2	2	2	2	4	4	4	4	4	5	8	8	8	8	8	···	8	10	10	10	10	16	···	16	16	···	16	20	20	20	20	20
size	1	1	2	2	2	1	1	2	2	2	4	5	5	5	5	10	···	10	4	8	8	8	5	···	5	10	···	10	4	4	8	8	8

50 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4	4	4	4	8
type	+	+	+	+						+	+	-	-
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	C₈	D₄○C₁₆	F₅	C₂×F₅	C₅⋊C₈	C₅⋊C₈	C₅⋊C₁₆.C₂²
kernel	C₅⋊C₁₆.C₂²	C₂×C₅⋊C₁₆	C₂₀.C₈	D₄.Dic₅	C₄.Dic₅	C₅×C₄○D₄	C₅×D₄	C₅×Q₈	C₅	C₄○D₄	C₂×C₄	D₄	Q₈	C₁
# reps	1	3	3	1	6	2	12	4	8	1	3	3	1	2

Matrix representation of C₅⋊C₁₆.C₂² ►in GL₆(𝔽₂₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	240	240	240	240

0	165	0	0	0	0
165	0	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162

1	0	0	0	0	0
0	240	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

165	0	0	0	0	0
0	165	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,165,0,0,0,0,165,0,0,0,0,0,0,0,79,129,160,139,0,0,218,104,239,27,0,0,106,23,137,2,0,0,81,102,25,162],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[165,0,0,0,0,0,0,165,0,0,0,0,0,0,79,129,160,139,0,0,218,104,239,27,0,0,106,23,137,2,0,0,81,102,25,162] >;

C₅⋊C₁₆.C₂² in GAP, Magma, Sage, TeX

C_5\rtimes C_{16}.C_2^2

% in TeX

G:=Group("C5:C16.C2^2");

// GroupNames label

G:=SmallGroup(320,1129);

// by ID

G=gap.SmallGroup(320,1129);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,80,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^16=c^2=1,d^2=b^2,b*a*b^-1=d*a*d^-1=a^3,a*c=c*a,c*b*c=b^9,b*d=d*b,c*d=d*c>;

// generators/relations

0	165	0	0	0	0
165	0	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162

165	0	0	0	0	0
0	165	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162

0	165	0	0	0	0
165	0	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162

165	0	0	0	0	0
0	165	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162

G = C5⋊C16.C22 order 320 = 26·5

The non-split extension by C5⋊C16 of C22 acting via C22/C2=C2

G = C₅⋊C₁₆.C₂² order 320 = 2⁶·5

The non-split extension by C₅⋊C₁₆ of C₂² acting via C₂²/C₂=C₂

0	165	0	0	0	0
165	0	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162

165	0	0	0	0	0
0	165	0	0	0	0
0	0	79	218	106	81
0	0	129	104	23	102
0	0	160	239	137	25
0	0	139	27	2	162