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## G = C4×Dic11order 176 = 24·11

### Direct product of C4 and Dic11

Aliases: C4×Dic11, C11⋊C42, C442C4, C22.3D22, C22.3(C2×C4), (C2×C44).7C2, (C2×C4).6D11, C2.2(C4×D11), (C2×C22).3C22, C2.2(C2×Dic11), (C2×Dic11).4C2, SmallGroup(176,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C4×Dic11
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C4×Dic11
 Lower central C11 — C4×Dic11
 Upper central C1 — C2×C4

Generators and relations for C4×Dic11
G = < a,b,c | a4=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C4×Dic11
Regular action on 176 points
Generators in S176
(1 105 145 171)(2 106 146 172)(3 107 147 173)(4 108 148 174)(5 109 149 175)(6 110 150 176)(7 89 151 155)(8 90 152 156)(9 91 153 157)(10 92 154 158)(11 93 133 159)(12 94 134 160)(13 95 135 161)(14 96 136 162)(15 97 137 163)(16 98 138 164)(17 99 139 165)(18 100 140 166)(19 101 141 167)(20 102 142 168)(21 103 143 169)(22 104 144 170)(23 120 67 65)(24 121 68 66)(25 122 69 45)(26 123 70 46)(27 124 71 47)(28 125 72 48)(29 126 73 49)(30 127 74 50)(31 128 75 51)(32 129 76 52)(33 130 77 53)(34 131 78 54)(35 132 79 55)(36 111 80 56)(37 112 81 57)(38 113 82 58)(39 114 83 59)(40 115 84 60)(41 116 85 61)(42 117 86 62)(43 118 87 63)(44 119 88 64)
(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)
(1 127 12 116)(2 126 13 115)(3 125 14 114)(4 124 15 113)(5 123 16 112)(6 122 17 111)(7 121 18 132)(8 120 19 131)(9 119 20 130)(10 118 21 129)(11 117 22 128)(23 167 34 156)(24 166 35 155)(25 165 36 176)(26 164 37 175)(27 163 38 174)(28 162 39 173)(29 161 40 172)(30 160 41 171)(31 159 42 170)(32 158 43 169)(33 157 44 168)(45 139 56 150)(46 138 57 149)(47 137 58 148)(48 136 59 147)(49 135 60 146)(50 134 61 145)(51 133 62 144)(52 154 63 143)(53 153 64 142)(54 152 65 141)(55 151 66 140)(67 101 78 90)(68 100 79 89)(69 99 80 110)(70 98 81 109)(71 97 82 108)(72 96 83 107)(73 95 84 106)(74 94 85 105)(75 93 86 104)(76 92 87 103)(77 91 88 102)

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

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

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

C4×Dic11 is a maximal subgroup of
Dic11⋊C8  C88⋊C4  D444C4  C44.56D4  C42×D11  C42⋊D11  C23.11D22  C23.D22  Dic114D4  Dic11.D4  Dic22⋊C4  C44⋊Q8  Dic11.Q8  C44.3Q8  C4⋊C47D11  D44⋊C4  C4⋊C4⋊D11  C23.21D22  C44.17D4  C44⋊D4  Dic11⋊Q8  C44.23D4
C4×Dic11 is a maximal quotient of
C42.D11  C88⋊C4  C22.C42

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4L 11A ··· 11E 22A ··· 22O 44A ··· 44T order 1 2 2 2 4 4 4 4 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 1 1 1 1 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 D11 Dic11 D22 C4×D11 kernel C4×Dic11 C2×Dic11 C2×C44 Dic11 C44 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 5 10 5 20

Matrix representation of C4×Dic11 in GL4(𝔽89) generated by

 34 0 0 0 0 55 0 0 0 0 88 0 0 0 0 88
,
 88 0 0 0 0 1 0 0 0 0 88 1 0 0 80 8
,
 34 0 0 0 0 88 0 0 0 0 65 13 0 0 79 24
G:=sub<GL(4,GF(89))| [34,0,0,0,0,55,0,0,0,0,88,0,0,0,0,88],[88,0,0,0,0,1,0,0,0,0,88,80,0,0,1,8],[34,0,0,0,0,88,0,0,0,0,65,79,0,0,13,24] >;

C4×Dic11 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{11}
% in TeX

G:=Group("C4xDic11");
// GroupNames label

G:=SmallGroup(176,10);
// by ID

G=gap.SmallGroup(176,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,20,46,4004]);
// Polycyclic

G:=Group<a,b,c|a^4=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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