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## G = C22×Dic11order 176 = 24·11

### Direct product of C22 and Dic11

Aliases: C22×Dic11, C22.9C23, C23.2D11, C22.11D22, (C2×C22)⋊3C4, C222(C2×C4), C112(C22×C4), (C22×C22).3C2, C2.2(C22×D11), (C2×C22).12C22, SmallGroup(176,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C22×Dic11
 Chief series C1 — C11 — C22 — Dic11 — C2×Dic11 — C22×Dic11
 Lower central C11 — C22×Dic11
 Upper central C1 — C23

Generators and relations for C22×Dic11
G = < a,b,c,d | a2=b2=c22=1, d2=c11, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 164 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C11, C22×C4, C22, C22, Dic11, C2×C22, C2×Dic11, C22×C22, C22×Dic11
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D11, Dic11, D22, C2×Dic11, C22×D11, C22×Dic11

Smallest permutation representation of C22×Dic11
Regular action on 176 points
Generators in S176
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 133)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 141)(109 142)(110 143)(111 166)(112 167)(113 168)(114 169)(115 170)(116 171)(117 172)(118 173)(119 174)(120 175)(121 176)(122 155)(123 156)(124 157)(125 158)(126 159)(127 160)(128 161)(129 162)(130 163)(131 164)(132 165)
(1 42)(2 43)(3 44)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 131)(99 132)(100 111)(101 112)(102 113)(103 114)(104 115)(105 116)(106 117)(107 118)(108 119)(109 120)(110 121)(133 166)(134 167)(135 168)(136 169)(137 170)(138 171)(139 172)(140 173)(141 174)(142 175)(143 176)(144 155)(145 156)(146 157)(147 158)(148 159)(149 160)(150 161)(151 162)(152 163)(153 164)(154 165)
(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)
(1 122 12 111)(2 121 13 132)(3 120 14 131)(4 119 15 130)(5 118 16 129)(6 117 17 128)(7 116 18 127)(8 115 19 126)(9 114 20 125)(10 113 21 124)(11 112 22 123)(23 108 34 97)(24 107 35 96)(25 106 36 95)(26 105 37 94)(27 104 38 93)(28 103 39 92)(29 102 40 91)(30 101 41 90)(31 100 42 89)(32 99 43 110)(33 98 44 109)(45 166 56 155)(46 165 57 176)(47 164 58 175)(48 163 59 174)(49 162 60 173)(50 161 61 172)(51 160 62 171)(52 159 63 170)(53 158 64 169)(54 157 65 168)(55 156 66 167)(67 148 78 137)(68 147 79 136)(69 146 80 135)(70 145 81 134)(71 144 82 133)(72 143 83 154)(73 142 84 153)(74 141 85 152)(75 140 86 151)(76 139 87 150)(77 138 88 149)

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

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

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

C22×Dic11 is a maximal subgroup of
C22.C42  C23.11D22  C22⋊Dic22  Dic114D4  C22.D44  C23.18D22  Dic11⋊D4  C22×C4×D11
C22×Dic11 is a maximal quotient of
C23.21D22  Q8.Dic11

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 11A ··· 11E 22A ··· 22AI order 1 2 ··· 2 4 ··· 4 11 ··· 11 22 ··· 22 size 1 1 ··· 1 11 ··· 11 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D11 Dic11 D22 kernel C22×Dic11 C2×Dic11 C22×C22 C2×C22 C23 C22 C22 # reps 1 6 1 8 5 20 15

Matrix representation of C22×Dic11 in GL4(𝔽89) generated by

 88 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
,
 1 0 0 0 0 88 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 87 0 0 0 0 44
,
 88 0 0 0 0 88 0 0 0 0 0 88 0 0 1 0
G:=sub<GL(4,GF(89))| [88,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,87,0,0,0,0,44],[88,0,0,0,0,88,0,0,0,0,0,1,0,0,88,0] >;

C22×Dic11 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{11}
% in TeX

G:=Group("C2^2xDic11");
// GroupNames label

G:=SmallGroup(176,35);
// by ID

G=gap.SmallGroup(176,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,40,4004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^22=1,d^2=c^11,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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