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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 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C11, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C22, C22 [×3], C4○D8, C44 [×2], C44 [×2], C2×C22, C2×C22 [×2], C88 [×2], C2×C44, C2×C44 [×2], D4×C11 [×2], D4×C11 [×2], Q8×C11 [×2], C2×C88, C11×D8, C11×SD16 [×2], C11×Q16, C11×C4○D4 [×2], C11×C4○D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C11, C2×D4, C22 [×7], C4○D8, C2×C22 [×7], D4×C11 [×2], 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 46 117)(2 91 47 118)(3 92 48 119)(4 93 49 120)(5 94 50 121)(6 95 51 111)(7 96 52 112)(8 97 53 113)(9 98 54 114)(10 99 55 115)(11 89 45 116)(12 163 29 136)(13 164 30 137)(14 165 31 138)(15 155 32 139)(16 156 33 140)(17 157 23 141)(18 158 24 142)(19 159 25 143)(20 160 26 133)(21 161 27 134)(22 162 28 135)(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)(56 80 76 100)(57 81 77 101)(58 82 67 102)(59 83 68 103)(60 84 69 104)(61 85 70 105)(62 86 71 106)(63 87 72 107)(64 88 73 108)(65 78 74 109)(66 79 75 110)
(1 161 75 154 46 134 66 123)(2 162 76 144 47 135 56 124)(3 163 77 145 48 136 57 125)(4 164 67 146 49 137 58 126)(5 165 68 147 50 138 59 127)(6 155 69 148 51 139 60 128)(7 156 70 149 52 140 61 129)(8 157 71 150 53 141 62 130)(9 158 72 151 54 142 63 131)(10 159 73 152 55 143 64 132)(11 160 74 153 45 133 65 122)(12 81 42 92 29 101 169 119)(13 82 43 93 30 102 170 120)(14 83 44 94 31 103 171 121)(15 84 34 95 32 104 172 111)(16 85 35 96 33 105 173 112)(17 86 36 97 23 106 174 113)(18 87 37 98 24 107 175 114)(19 88 38 99 25 108 176 115)(20 78 39 89 26 109 166 116)(21 79 40 90 27 110 167 117)(22 80 41 91 28 100 168 118)
(12 169)(13 170)(14 171)(15 172)(16 173)(17 174)(18 175)(19 176)(20 166)(21 167)(22 168)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 34)(33 35)(56 76)(57 77)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)(66 75)(78 109)(79 110)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(122 160)(123 161)(124 162)(125 163)(126 164)(127 165)(128 155)(129 156)(130 157)(131 158)(132 159)(133 153)(134 154)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)(141 150)(142 151)(143 152)

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

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

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

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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