Copied to
clipboard

## G = C11×SD32order 352 = 25·11

### Direct product of C11 and SD32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C11×SD32, D8.C22, C162C22, C1766C2, Q161C22, C22.16D8, C44.37D4, C88.25C22, C8.3(C2×C22), C4.2(D4×C11), C2.4(C11×D8), (C11×Q16)⋊5C2, (C11×D8).2C2, SmallGroup(352,61)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C11×SD32
 Chief series C1 — C2 — C4 — C8 — C88 — C11×Q16 — C11×SD32
 Lower central C1 — C2 — C4 — C8 — C11×SD32
 Upper central C1 — C22 — C44 — C88 — C11×SD32

Generators and relations for C11×SD32
G = < a,b,c | a11=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Smallest permutation representation of C11×SD32
On 176 points
Generators in S176
(1 120 95 37 146 63 29 112 163 65 142)(2 121 96 38 147 64 30 97 164 66 143)(3 122 81 39 148 49 31 98 165 67 144)(4 123 82 40 149 50 32 99 166 68 129)(5 124 83 41 150 51 17 100 167 69 130)(6 125 84 42 151 52 18 101 168 70 131)(7 126 85 43 152 53 19 102 169 71 132)(8 127 86 44 153 54 20 103 170 72 133)(9 128 87 45 154 55 21 104 171 73 134)(10 113 88 46 155 56 22 105 172 74 135)(11 114 89 47 156 57 23 106 173 75 136)(12 115 90 48 157 58 24 107 174 76 137)(13 116 91 33 158 59 25 108 175 77 138)(14 117 92 34 159 60 26 109 176 78 139)(15 118 93 35 160 61 27 110 161 79 140)(16 119 94 36 145 62 28 111 162 80 141)
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 61)(50 52)(51 59)(53 57)(54 64)(56 62)(58 60)(66 72)(67 79)(68 70)(69 77)(71 75)(74 80)(76 78)(81 93)(82 84)(83 91)(85 89)(86 96)(88 94)(90 92)(97 103)(98 110)(99 101)(100 108)(102 106)(105 111)(107 109)(113 119)(114 126)(115 117)(116 124)(118 122)(121 127)(123 125)(129 131)(130 138)(132 136)(133 143)(135 141)(137 139)(140 144)(145 155)(147 153)(148 160)(149 151)(150 158)(152 156)(157 159)(161 165)(162 172)(164 170)(166 168)(167 175)(169 173)(174 176)

G:=sub<Sym(176)| (1,120,95,37,146,63,29,112,163,65,142)(2,121,96,38,147,64,30,97,164,66,143)(3,122,81,39,148,49,31,98,165,67,144)(4,123,82,40,149,50,32,99,166,68,129)(5,124,83,41,150,51,17,100,167,69,130)(6,125,84,42,151,52,18,101,168,70,131)(7,126,85,43,152,53,19,102,169,71,132)(8,127,86,44,153,54,20,103,170,72,133)(9,128,87,45,154,55,21,104,171,73,134)(10,113,88,46,155,56,22,105,172,74,135)(11,114,89,47,156,57,23,106,173,75,136)(12,115,90,48,157,58,24,107,174,76,137)(13,116,91,33,158,59,25,108,175,77,138)(14,117,92,34,159,60,26,109,176,78,139)(15,118,93,35,160,61,27,110,161,79,140)(16,119,94,36,145,62,28,111,162,80,141), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(66,72)(67,79)(68,70)(69,77)(71,75)(74,80)(76,78)(81,93)(82,84)(83,91)(85,89)(86,96)(88,94)(90,92)(97,103)(98,110)(99,101)(100,108)(102,106)(105,111)(107,109)(113,119)(114,126)(115,117)(116,124)(118,122)(121,127)(123,125)(129,131)(130,138)(132,136)(133,143)(135,141)(137,139)(140,144)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159)(161,165)(162,172)(164,170)(166,168)(167,175)(169,173)(174,176)>;

G:=Group( (1,120,95,37,146,63,29,112,163,65,142)(2,121,96,38,147,64,30,97,164,66,143)(3,122,81,39,148,49,31,98,165,67,144)(4,123,82,40,149,50,32,99,166,68,129)(5,124,83,41,150,51,17,100,167,69,130)(6,125,84,42,151,52,18,101,168,70,131)(7,126,85,43,152,53,19,102,169,71,132)(8,127,86,44,153,54,20,103,170,72,133)(9,128,87,45,154,55,21,104,171,73,134)(10,113,88,46,155,56,22,105,172,74,135)(11,114,89,47,156,57,23,106,173,75,136)(12,115,90,48,157,58,24,107,174,76,137)(13,116,91,33,158,59,25,108,175,77,138)(14,117,92,34,159,60,26,109,176,78,139)(15,118,93,35,160,61,27,110,161,79,140)(16,119,94,36,145,62,28,111,162,80,141), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(66,72)(67,79)(68,70)(69,77)(71,75)(74,80)(76,78)(81,93)(82,84)(83,91)(85,89)(86,96)(88,94)(90,92)(97,103)(98,110)(99,101)(100,108)(102,106)(105,111)(107,109)(113,119)(114,126)(115,117)(116,124)(118,122)(121,127)(123,125)(129,131)(130,138)(132,136)(133,143)(135,141)(137,139)(140,144)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159)(161,165)(162,172)(164,170)(166,168)(167,175)(169,173)(174,176) );

G=PermutationGroup([(1,120,95,37,146,63,29,112,163,65,142),(2,121,96,38,147,64,30,97,164,66,143),(3,122,81,39,148,49,31,98,165,67,144),(4,123,82,40,149,50,32,99,166,68,129),(5,124,83,41,150,51,17,100,167,69,130),(6,125,84,42,151,52,18,101,168,70,131),(7,126,85,43,152,53,19,102,169,71,132),(8,127,86,44,153,54,20,103,170,72,133),(9,128,87,45,154,55,21,104,171,73,134),(10,113,88,46,155,56,22,105,172,74,135),(11,114,89,47,156,57,23,106,173,75,136),(12,115,90,48,157,58,24,107,174,76,137),(13,116,91,33,158,59,25,108,175,77,138),(14,117,92,34,159,60,26,109,176,78,139),(15,118,93,35,160,61,27,110,161,79,140),(16,119,94,36,145,62,28,111,162,80,141)], [(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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(33,41),(34,48),(35,39),(36,46),(38,44),(40,42),(43,47),(49,61),(50,52),(51,59),(53,57),(54,64),(56,62),(58,60),(66,72),(67,79),(68,70),(69,77),(71,75),(74,80),(76,78),(81,93),(82,84),(83,91),(85,89),(86,96),(88,94),(90,92),(97,103),(98,110),(99,101),(100,108),(102,106),(105,111),(107,109),(113,119),(114,126),(115,117),(116,124),(118,122),(121,127),(123,125),(129,131),(130,138),(132,136),(133,143),(135,141),(137,139),(140,144),(145,155),(147,153),(148,160),(149,151),(150,158),(152,156),(157,159),(161,165),(162,172),(164,170),(166,168),(167,175),(169,173),(174,176)])

121 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 11A ··· 11J 16A 16B 16C 16D 22A ··· 22J 22K ··· 22T 44A ··· 44J 44K ··· 44T 88A ··· 88T 176A ··· 176AN order 1 2 2 4 4 8 8 11 ··· 11 16 16 16 16 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 176 ··· 176 size 1 1 8 2 8 2 2 1 ··· 1 2 2 2 2 1 ··· 1 8 ··· 8 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2

121 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C11 C22 C22 C22 D4 D8 SD32 D4×C11 C11×D8 C11×SD32 kernel C11×SD32 C176 C11×D8 C11×Q16 SD32 C16 D8 Q16 C44 C22 C11 C4 C2 C1 # reps 1 1 1 1 10 10 10 10 1 2 4 10 20 40

Matrix representation of C11×SD32 in GL2(𝔽23) generated by

 9 0 0 9
,
 16 15 20 0
,
 22 10 0 1
G:=sub<GL(2,GF(23))| [9,0,0,9],[16,20,15,0],[22,0,10,1] >;

C11×SD32 in GAP, Magma, Sage, TeX

C_{11}\times {\rm SD}_{32}
% in TeX

G:=Group("C11xSD32");
// GroupNames label

G:=SmallGroup(352,61);
// by ID

G=gap.SmallGroup(352,61);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,1056,553,3171,1593,165,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c|a^11=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations

Export

׿
×
𝔽