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## G = C11×C8.C22order 352 = 25·11

### Direct product of C11 and C8.C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C11×C8.C22
 Chief series C1 — C2 — C4 — C44 — D4×C11 — C11×SD16 — C11×C8.C22
 Lower central C1 — C2 — C4 — C11×C8.C22
 Upper central C1 — C22 — C2×C44 — C11×C8.C22

Generators and relations for C11×C8.C22
G = < a,b,c,d | a11=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C11, M4(2), SD16, Q16, C2×Q8, C4○D4, C22, C22, C8.C22, C44, C44, C2×C22, C2×C22, C88, C2×C44, C2×C44, D4×C11, D4×C11, Q8×C11, Q8×C11, Q8×C11, C11×M4(2), C11×SD16, C11×Q16, Q8×C22, C11×C4○D4, C11×C8.C22
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C22, C8.C22, C2×C22, D4×C11, C22×C22, D4×C22, C11×C8.C22

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

121 conjugacy classes

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

121 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C11 C22 C22 C22 C22 C22 D4 D4 D4×C11 D4×C11 C8.C22 C11×C8.C22 kernel C11×C8.C22 C11×M4(2) C11×SD16 C11×Q16 Q8×C22 C11×C4○D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C44 C2×C22 C4 C22 C11 C1 # reps 1 1 2 2 1 1 10 10 20 20 10 10 1 1 10 10 1 10

Matrix representation of C11×C8.C22 in GL4(𝔽89) generated by

 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
,
 41 54 3 86 60 20 38 1 55 55 46 58 34 55 52 71
,
 1 0 63 0 0 88 0 64 0 0 88 0 0 0 0 1
,
 63 2 66 88 40 25 69 85 87 0 26 87 0 87 49 64
G:=sub<GL(4,GF(89))| [32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[41,60,55,34,54,20,55,55,3,38,46,52,86,1,58,71],[1,0,0,0,0,88,0,0,63,0,88,0,0,64,0,1],[63,40,87,0,2,25,0,87,66,69,26,49,88,85,87,64] >;

C11×C8.C22 in GAP, Magma, Sage, TeX

C_{11}\times C_8.C_2^2
% in TeX

G:=Group("C11xC8.C2^2");
// GroupNames label

G:=SmallGroup(352,172);
// by ID

G=gap.SmallGroup(352,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,1063,3242,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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