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## G = C11×C4○D8order 352 = 25·11

### Direct product of C11 and C4○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C11×C4○D8
 Chief series C1 — C2 — C4 — C44 — D4×C11 — C11×D8 — C11×C4○D8
 Lower central C1 — C2 — C4 — C11×C4○D8
 Upper central C1 — C44 — C2×C44 — C11×C4○D8

Generators and relations for C11×C4○D8
G = < a,b,c,d | a11=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C11, C2×C8, D8, SD16, Q16, C4○D4, C22, C22, C4○D8, C44, C44, C2×C22, C2×C22, C88, C2×C44, C2×C44, D4×C11, D4×C11, Q8×C11, C2×C88, C11×D8, C11×SD16, C11×Q16, C11×C4○D4, C11×C4○D8
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C22, C4○D8, C2×C22, D4×C11, C22×C22, D4×C22, C11×C4○D8

Smallest permutation representation of C11×C4○D8
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 90 73 117)(2 91 74 118)(3 92 75 119)(4 93 76 120)(5 94 77 121)(6 95 67 111)(7 96 68 112)(8 97 69 113)(9 98 70 114)(10 99 71 115)(11 89 72 116)(12 155 32 139)(13 156 33 140)(14 157 23 141)(15 158 24 142)(16 159 25 143)(17 160 26 133)(18 161 27 134)(19 162 28 135)(20 163 29 136)(21 164 30 137)(22 165 31 138)(34 145 172 128)(35 146 173 129)(36 147 174 130)(37 148 175 131)(38 149 176 132)(39 150 166 122)(40 151 167 123)(41 152 168 124)(42 153 169 125)(43 154 170 126)(44 144 171 127)(45 101 57 84)(46 102 58 85)(47 103 59 86)(48 104 60 87)(49 105 61 88)(50 106 62 78)(51 107 63 79)(52 108 64 80)(53 109 65 81)(54 110 66 82)(55 100 56 83)
(1 161 51 151 73 134 63 123)(2 162 52 152 74 135 64 124)(3 163 53 153 75 136 65 125)(4 164 54 154 76 137 66 126)(5 165 55 144 77 138 56 127)(6 155 45 145 67 139 57 128)(7 156 46 146 68 140 58 129)(8 157 47 147 69 141 59 130)(9 158 48 148 70 142 60 131)(10 159 49 149 71 143 61 132)(11 160 50 150 72 133 62 122)(12 84 34 95 32 101 172 111)(13 85 35 96 33 102 173 112)(14 86 36 97 23 103 174 113)(15 87 37 98 24 104 175 114)(16 88 38 99 25 105 176 115)(17 78 39 89 26 106 166 116)(18 79 40 90 27 107 167 117)(19 80 41 91 28 108 168 118)(20 81 42 92 29 109 169 119)(21 82 43 93 30 110 170 120)(22 83 44 94 31 100 171 121)
(12 172)(13 173)(14 174)(15 175)(16 176)(17 166)(18 167)(19 168)(20 169)(21 170)(22 171)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 34)(33 35)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 56)(78 106)(79 107)(80 108)(81 109)(82 110)(83 100)(84 101)(85 102)(86 103)(87 104)(88 105)(122 160)(123 161)(124 162)(125 163)(126 164)(127 165)(128 155)(129 156)(130 157)(131 158)(132 159)(133 150)(134 151)(135 152)(136 153)(137 154)(138 144)(139 145)(140 146)(141 147)(142 148)(143 149)

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

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

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

154 conjugacy classes

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

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C11 C22 C22 C22 C22 C22 D4 D4 C4○D8 D4×C11 D4×C11 C11×C4○D8 kernel C11×C4○D8 C2×C88 C11×D8 C11×SD16 C11×Q16 C11×C4○D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C44 C2×C22 C11 C4 C22 C1 # reps 1 1 1 2 1 2 10 10 10 20 10 20 1 1 4 10 10 40

Matrix representation of C11×C4○D8 in GL2(𝔽89) generated by

 16 0 0 16
,
 34 0 0 34
,
 0 25 32 25
,
 1 0 1 88
G:=sub<GL(2,GF(89))| [16,0,0,16],[34,0,0,34],[0,32,25,25],[1,1,0,88] >;

C11×C4○D8 in GAP, Magma, Sage, TeX

C_{11}\times C_4\circ D_8
% in TeX

G:=Group("C11xC4oD8");
// GroupNames label

G:=SmallGroup(352,170);
// by ID

G=gap.SmallGroup(352,170);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,806,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations

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