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## G = C4⋊C4×D11order 352 = 25·11

### Direct product of C4⋊C4 and D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C4⋊C4×D11
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×C4×D11 — C4⋊C4×D11
 Lower central C11 — C22 — C4⋊C4×D11
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊C4×D11
G = < a,b,c,d | a4=b4=c11=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 498 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, C23, C11, C4⋊C4, C4⋊C4, C22×C4, D11, C22, C2×C4⋊C4, Dic11, Dic11, C44, C44, D22, C2×C22, C4×D11, C4×D11, C2×Dic11, C2×Dic11, C2×C44, C2×C44, C22×D11, Dic11⋊C4, C44⋊C4, C11×C4⋊C4, C2×C4×D11, C2×C4×D11, C4⋊C4×D11
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, D11, C2×C4⋊C4, D22, C4×D11, C22×D11, C2×C4×D11, D4×D11, Q8×D11, C4⋊C4×D11

Smallest permutation representation of C4⋊C4×D11
On 176 points
Generators in S176
(1 131 21 120)(2 132 22 121)(3 122 12 111)(4 123 13 112)(5 124 14 113)(6 125 15 114)(7 126 16 115)(8 127 17 116)(9 128 18 117)(10 129 19 118)(11 130 20 119)(23 100 34 89)(24 101 35 90)(25 102 36 91)(26 103 37 92)(27 104 38 93)(28 105 39 94)(29 106 40 95)(30 107 41 96)(31 108 42 97)(32 109 43 98)(33 110 44 99)(45 166 56 155)(46 167 57 156)(47 168 58 157)(48 169 59 158)(49 170 60 159)(50 171 61 160)(51 172 62 161)(52 173 63 162)(53 174 64 163)(54 175 65 164)(55 176 66 165)(67 144 78 133)(68 145 79 134)(69 146 80 135)(70 147 81 136)(71 148 82 137)(72 149 83 138)(73 150 84 139)(74 151 85 140)(75 152 86 141)(76 153 87 142)(77 154 88 143)
(1 76 32 54)(2 77 33 55)(3 67 23 45)(4 68 24 46)(5 69 25 47)(6 70 26 48)(7 71 27 49)(8 72 28 50)(9 73 29 51)(10 74 30 52)(11 75 31 53)(12 78 34 56)(13 79 35 57)(14 80 36 58)(15 81 37 59)(16 82 38 60)(17 83 39 61)(18 84 40 62)(19 85 41 63)(20 86 42 64)(21 87 43 65)(22 88 44 66)(89 166 111 144)(90 167 112 145)(91 168 113 146)(92 169 114 147)(93 170 115 148)(94 171 116 149)(95 172 117 150)(96 173 118 151)(97 174 119 152)(98 175 120 153)(99 176 121 154)(100 155 122 133)(101 156 123 134)(102 157 124 135)(103 158 125 136)(104 159 126 137)(105 160 127 138)(106 161 128 139)(107 162 129 140)(108 163 130 141)(109 164 131 142)(110 165 132 143)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 110)(97 109)(98 108)(99 107)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 132)(119 131)(120 130)(121 129)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 154)(141 153)(142 152)(143 151)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 176)(163 175)(164 174)(165 173)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 D4 Q8 D11 D22 C4×D11 D4×D11 Q8×D11 kernel C4⋊C4×D11 Dic11⋊C4 C44⋊C4 C11×C4⋊C4 C2×C4×D11 C4×D11 D22 D22 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 2 1 1 3 8 2 2 5 15 20 5 5

Matrix representation of C4⋊C4×D11 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 1 2 0 0 88 88
,
 55 0 0 0 0 55 0 0 0 0 50 2 0 0 40 39
,
 37 1 0 0 42 18 0 0 0 0 1 0 0 0 0 1
,
 6 84 0 0 7 83 0 0 0 0 88 0 0 0 0 88
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,88,0,0,2,88],[55,0,0,0,0,55,0,0,0,0,50,40,0,0,2,39],[37,42,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[6,7,0,0,84,83,0,0,0,0,88,0,0,0,0,88] >;

C4⋊C4×D11 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times D_{11}
% in TeX

G:=Group("C4:C4xD11");
// GroupNames label

G:=SmallGroup(352,86);
// by ID

G=gap.SmallGroup(352,86);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,188,50,11525]);
// Polycyclic

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

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