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, Q8.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
Subgroups: 178 in 84 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C16 [×4], C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C2×C16 [×3], M5(2) [×3], C8○D4, C5⋊2C8, C5⋊2C8 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, D4○C16, C5⋊C16, C5⋊C16 [×3], C2×C5⋊2C8 [×3], C4.Dic5 [×3], C5×C4○D4, C2×C5⋊C16 [×3], C20.C8 [×3], Q8.Dic5, C5⋊C16.C22
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, C22×C8, C5⋊C8 [×4], C2×F5 [×3], D4○C16, C2×C5⋊C8 [×6], C22×F5, C22×C5⋊C8, C5⋊C16.C22
Generators and relations
G = < a,b,c,d | a5=b16=c2=1, d2=b2, bab-1=dad-1=a3, ac=ca, cbc=b9, bd=db, cd=dc >
►On 160 points
Generators in S160
(1 115 28 33 111)(2 34 116 112 29)(3 97 35 30 117)(4 31 98 118 36)(5 119 32 37 99)(6 38 120 100 17)(7 101 39 18 121)(8 19 102 122 40)(9 123 20 41 103)(10 42 124 104 21)(11 105 43 22 125)(12 23 106 126 44)(13 127 24 45 107)(14 46 128 108 25)(15 109 47 26 113)(16 27 110 114 48)(49 134 77 82 150)(50 83 135 151 78)(51 152 84 79 136)(52 80 153 137 85)(53 138 65 86 154)(54 87 139 155 66)(55 156 88 67 140)(56 68 157 141 89)(57 142 69 90 158)(58 91 143 159 70)(59 160 92 71 144)(60 72 145 129 93)(61 130 73 94 146)(62 95 131 147 74)(63 148 96 75 132)(64 76 149 133 81)
(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)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(82 90)(84 92)(86 94)(88 96)(98 106)(100 108)(102 110)(104 112)(114 122)(116 124)(118 126)(120 128)(130 138)(132 140)(134 142)(136 144)(146 154)(148 156)(150 158)(152 160)
(1 66 3 68 5 70 7 72 9 74 11 76 13 78 15 80)(2 67 4 69 6 71 8 73 10 75 12 77 14 79 16 65)(17 160 19 146 21 148 23 150 25 152 27 154 29 156 31 158)(18 145 20 147 22 149 24 151 26 153 28 155 30 157 32 159)(33 54 35 56 37 58 39 60 41 62 43 64 45 50 47 52)(34 55 36 57 38 59 40 61 42 63 44 49 46 51 48 53)(81 127 83 113 85 115 87 117 89 119 91 121 93 123 95 125)(82 128 84 114 86 116 88 118 90 120 92 122 94 124 96 126)(97 141 99 143 101 129 103 131 105 133 107 135 109 137 111 139)(98 142 100 144 102 130 104 132 106 134 108 136 110 138 112 140)
G:=sub<Sym(160)| (1,115,28,33,111)(2,34,116,112,29)(3,97,35,30,117)(4,31,98,118,36)(5,119,32,37,99)(6,38,120,100,17)(7,101,39,18,121)(8,19,102,122,40)(9,123,20,41,103)(10,42,124,104,21)(11,105,43,22,125)(12,23,106,126,44)(13,127,24,45,107)(14,46,128,108,25)(15,109,47,26,113)(16,27,110,114,48)(49,134,77,82,150)(50,83,135,151,78)(51,152,84,79,136)(52,80,153,137,85)(53,138,65,86,154)(54,87,139,155,66)(55,156,88,67,140)(56,68,157,141,89)(57,142,69,90,158)(58,91,143,159,70)(59,160,92,71,144)(60,72,145,129,93)(61,130,73,94,146)(62,95,131,147,74)(63,148,96,75,132)(64,76,149,133,81), (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)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(82,90)(84,92)(86,94)(88,96)(98,106)(100,108)(102,110)(104,112)(114,122)(116,124)(118,126)(120,128)(130,138)(132,140)(134,142)(136,144)(146,154)(148,156)(150,158)(152,160), (1,66,3,68,5,70,7,72,9,74,11,76,13,78,15,80)(2,67,4,69,6,71,8,73,10,75,12,77,14,79,16,65)(17,160,19,146,21,148,23,150,25,152,27,154,29,156,31,158)(18,145,20,147,22,149,24,151,26,153,28,155,30,157,32,159)(33,54,35,56,37,58,39,60,41,62,43,64,45,50,47,52)(34,55,36,57,38,59,40,61,42,63,44,49,46,51,48,53)(81,127,83,113,85,115,87,117,89,119,91,121,93,123,95,125)(82,128,84,114,86,116,88,118,90,120,92,122,94,124,96,126)(97,141,99,143,101,129,103,131,105,133,107,135,109,137,111,139)(98,142,100,144,102,130,104,132,106,134,108,136,110,138,112,140)>;
G:=Group( (1,115,28,33,111)(2,34,116,112,29)(3,97,35,30,117)(4,31,98,118,36)(5,119,32,37,99)(6,38,120,100,17)(7,101,39,18,121)(8,19,102,122,40)(9,123,20,41,103)(10,42,124,104,21)(11,105,43,22,125)(12,23,106,126,44)(13,127,24,45,107)(14,46,128,108,25)(15,109,47,26,113)(16,27,110,114,48)(49,134,77,82,150)(50,83,135,151,78)(51,152,84,79,136)(52,80,153,137,85)(53,138,65,86,154)(54,87,139,155,66)(55,156,88,67,140)(56,68,157,141,89)(57,142,69,90,158)(58,91,143,159,70)(59,160,92,71,144)(60,72,145,129,93)(61,130,73,94,146)(62,95,131,147,74)(63,148,96,75,132)(64,76,149,133,81), (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)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(82,90)(84,92)(86,94)(88,96)(98,106)(100,108)(102,110)(104,112)(114,122)(116,124)(118,126)(120,128)(130,138)(132,140)(134,142)(136,144)(146,154)(148,156)(150,158)(152,160), (1,66,3,68,5,70,7,72,9,74,11,76,13,78,15,80)(2,67,4,69,6,71,8,73,10,75,12,77,14,79,16,65)(17,160,19,146,21,148,23,150,25,152,27,154,29,156,31,158)(18,145,20,147,22,149,24,151,26,153,28,155,30,157,32,159)(33,54,35,56,37,58,39,60,41,62,43,64,45,50,47,52)(34,55,36,57,38,59,40,61,42,63,44,49,46,51,48,53)(81,127,83,113,85,115,87,117,89,119,91,121,93,123,95,125)(82,128,84,114,86,116,88,118,90,120,92,122,94,124,96,126)(97,141,99,143,101,129,103,131,105,133,107,135,109,137,111,139)(98,142,100,144,102,130,104,132,106,134,108,136,110,138,112,140) );
G=PermutationGroup([(1,115,28,33,111),(2,34,116,112,29),(3,97,35,30,117),(4,31,98,118,36),(5,119,32,37,99),(6,38,120,100,17),(7,101,39,18,121),(8,19,102,122,40),(9,123,20,41,103),(10,42,124,104,21),(11,105,43,22,125),(12,23,106,126,44),(13,127,24,45,107),(14,46,128,108,25),(15,109,47,26,113),(16,27,110,114,48),(49,134,77,82,150),(50,83,135,151,78),(51,152,84,79,136),(52,80,153,137,85),(53,138,65,86,154),(54,87,139,155,66),(55,156,88,67,140),(56,68,157,141,89),(57,142,69,90,158),(58,91,143,159,70),(59,160,92,71,144),(60,72,145,129,93),(61,130,73,94,146),(62,95,131,147,74),(63,148,96,75,132),(64,76,149,133,81)], [(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),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(65,73),(67,75),(69,77),(71,79),(82,90),(84,92),(86,94),(88,96),(98,106),(100,108),(102,110),(104,112),(114,122),(116,124),(118,126),(120,128),(130,138),(132,140),(134,142),(136,144),(146,154),(148,156),(150,158),(152,160)], [(1,66,3,68,5,70,7,72,9,74,11,76,13,78,15,80),(2,67,4,69,6,71,8,73,10,75,12,77,14,79,16,65),(17,160,19,146,21,148,23,150,25,152,27,154,29,156,31,158),(18,145,20,147,22,149,24,151,26,153,28,155,30,157,32,159),(33,54,35,56,37,58,39,60,41,62,43,64,45,50,47,52),(34,55,36,57,38,59,40,61,42,63,44,49,46,51,48,53),(81,127,83,113,85,115,87,117,89,119,91,121,93,123,95,125),(82,128,84,114,86,116,88,118,90,120,92,122,94,124,96,126),(97,141,99,143,101,129,103,131,105,133,107,135,109,137,111,139),(98,142,100,144,102,130,104,132,106,134,108,136,110,138,112,140)])
Matrix representation ►G ⊆ 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] >;
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 | Q8.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 |
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
×
⋊
ℤ
𝔽
○
≀