Copied to
clipboard

## G = C11×C8○D4order 352 = 25·11

### Direct product of C11 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×C8○D4
 Chief series C1 — C2 — C4 — C44 — C88 — C2×C88 — C11×C8○D4
 Lower central C1 — C2 — C11×C8○D4
 Upper central C1 — C88 — C11×C8○D4

Generators and relations for C11×C8○D4
G = < a,b,c,d | a11=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C11, C2×C8 [×3], M4(2) [×3], C4○D4, C22, C22 [×3], C8○D4, C44, C44 [×3], C2×C22 [×3], C88, C88 [×3], C2×C44 [×3], D4×C11 [×3], Q8×C11, C2×C88 [×3], C11×M4(2) [×3], C11×C4○D4, C11×C8○D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C11, C22×C4, C22 [×7], C8○D4, C44 [×4], C2×C22 [×7], C2×C44 [×6], C22×C22, C22×C44, C11×C8○D4

Smallest permutation representation of C11×C8○D4
On 176 points
Generators in S176
(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)
(1 154 66 161 73 123 51 137)(2 144 56 162 74 124 52 138)(3 145 57 163 75 125 53 139)(4 146 58 164 76 126 54 140)(5 147 59 165 77 127 55 141)(6 148 60 155 67 128 45 142)(7 149 61 156 68 129 46 143)(8 150 62 157 69 130 47 133)(9 151 63 158 70 131 48 134)(10 152 64 159 71 132 49 135)(11 153 65 160 72 122 50 136)(12 95 42 101 32 119 169 81)(13 96 43 102 33 120 170 82)(14 97 44 103 23 121 171 83)(15 98 34 104 24 111 172 84)(16 99 35 105 25 112 173 85)(17 89 36 106 26 113 174 86)(18 90 37 107 27 114 175 87)(19 91 38 108 28 115 176 88)(20 92 39 109 29 116 166 78)(21 93 40 110 30 117 167 79)(22 94 41 100 31 118 168 80)
(1 117 73 93)(2 118 74 94)(3 119 75 95)(4 120 76 96)(5 121 77 97)(6 111 67 98)(7 112 68 99)(8 113 69 89)(9 114 70 90)(10 115 71 91)(11 116 72 92)(12 139 32 163)(13 140 33 164)(14 141 23 165)(15 142 24 155)(16 143 25 156)(17 133 26 157)(18 134 27 158)(19 135 28 159)(20 136 29 160)(21 137 30 161)(22 138 31 162)(34 148 172 128)(35 149 173 129)(36 150 174 130)(37 151 175 131)(38 152 176 132)(39 153 166 122)(40 154 167 123)(41 144 168 124)(42 145 169 125)(43 146 170 126)(44 147 171 127)(45 104 60 84)(46 105 61 85)(47 106 62 86)(48 107 63 87)(49 108 64 88)(50 109 65 78)(51 110 66 79)(52 100 56 80)(53 101 57 81)(54 102 58 82)(55 103 59 83)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 89)(9 90)(10 91)(11 92)(12 139)(13 140)(14 141)(15 142)(16 143)(17 133)(18 134)(19 135)(20 136)(21 137)(22 138)(23 165)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 161)(31 162)(32 163)(33 164)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 144)(42 145)(43 146)(44 147)(45 84)(46 85)(47 86)(48 87)(49 88)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 121)(122 166)(123 167)(124 168)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)

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

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

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

220 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E ··· 8J 11A ··· 11J 22A ··· 22J 22K ··· 22AN 44A ··· 44T 44U ··· 44AX 88A ··· 88AN 88AO ··· 88CV order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 8 ··· 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 88 ··· 88 size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

220 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C11 C22 C22 C22 C44 C44 C8○D4 C11×C8○D4 kernel C11×C8○D4 C2×C88 C11×M4(2) C11×C4○D4 D4×C11 Q8×C11 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C11 C1 # reps 1 3 3 1 6 2 10 30 30 10 60 20 4 40

Matrix representation of C11×C8○D4 in GL2(𝔽89) generated by

 8 0 0 8
,
 52 0 0 52
,
 0 1 88 0
,
 0 1 1 0
G:=sub<GL(2,GF(89))| [8,0,0,8],[52,0,0,52],[0,88,1,0],[0,1,1,0] >;

C11×C8○D4 in GAP, Magma, Sage, TeX

C_{11}\times C_8\circ D_4
% in TeX

G:=Group("C11xC8oD4");
// GroupNames label

G:=SmallGroup(352,166);
// by ID

G=gap.SmallGroup(352,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,528,1628,88]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

׿
×
𝔽