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## G = D4×Dic11order 352 = 25·11

### Direct product of D4 and Dic11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — D4×Dic11
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C22×Dic11 — D4×Dic11
 Lower central C11 — C22 — D4×Dic11
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×Dic11
G = < a,b,c,d | a4=b2=c22=1, d2=c11, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 394 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C11, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C22, C22, C4×D4, Dic11, Dic11, C44, C2×C22, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C2×Dic11, C2×C44, D4×C11, C22×C22, C4×Dic11, C44⋊C4, C23.D11, C22×Dic11, D4×C22, D4×Dic11
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D11, C4×D4, Dic11, D22, C2×Dic11, C22×D11, D4×D11, D42D11, C22×Dic11, D4×Dic11

Smallest permutation representation of D4×Dic11
On 176 points
Generators in S176
(1 60 168 111)(2 61 169 112)(3 62 170 113)(4 63 171 114)(5 64 172 115)(6 65 173 116)(7 66 174 117)(8 45 175 118)(9 46 176 119)(10 47 155 120)(11 48 156 121)(12 49 157 122)(13 50 158 123)(14 51 159 124)(15 52 160 125)(16 53 161 126)(17 54 162 127)(18 55 163 128)(19 56 164 129)(20 57 165 130)(21 58 166 131)(22 59 167 132)(23 70 139 110)(24 71 140 89)(25 72 141 90)(26 73 142 91)(27 74 143 92)(28 75 144 93)(29 76 145 94)(30 77 146 95)(31 78 147 96)(32 79 148 97)(33 80 149 98)(34 81 150 99)(35 82 151 100)(36 83 152 101)(37 84 153 102)(38 85 154 103)(39 86 133 104)(40 87 134 105)(41 88 135 106)(42 67 136 107)(43 68 137 108)(44 69 138 109)
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 110)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 175)(46 176)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)(85 154)(86 133)(87 134)(88 135)
(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)
(1 145 12 134)(2 144 13 133)(3 143 14 154)(4 142 15 153)(5 141 16 152)(6 140 17 151)(7 139 18 150)(8 138 19 149)(9 137 20 148)(10 136 21 147)(11 135 22 146)(23 163 34 174)(24 162 35 173)(25 161 36 172)(26 160 37 171)(27 159 38 170)(28 158 39 169)(29 157 40 168)(30 156 41 167)(31 155 42 166)(32 176 43 165)(33 175 44 164)(45 109 56 98)(46 108 57 97)(47 107 58 96)(48 106 59 95)(49 105 60 94)(50 104 61 93)(51 103 62 92)(52 102 63 91)(53 101 64 90)(54 100 65 89)(55 99 66 110)(67 131 78 120)(68 130 79 119)(69 129 80 118)(70 128 81 117)(71 127 82 116)(72 126 83 115)(73 125 84 114)(74 124 85 113)(75 123 86 112)(76 122 87 111)(77 121 88 132)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 D11 D22 Dic11 D22 D4×D11 D4⋊2D11 kernel D4×Dic11 C4×Dic11 C44⋊C4 C23.D11 C22×Dic11 D4×C22 D4×C11 Dic11 C22 C2×D4 C2×C4 D4 C23 C2 C2 # reps 1 1 1 2 2 1 8 2 2 5 5 20 10 5 5

Matrix representation of D4×Dic11 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 88 87 0 0 1 1
,
 88 0 0 0 0 88 0 0 0 0 88 87 0 0 0 1
,
 72 88 0 0 74 20 0 0 0 0 1 0 0 0 0 1
,
 9 5 0 0 37 80 0 0 0 0 88 0 0 0 0 88
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,1,0,0,87,1],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,87,1],[72,74,0,0,88,20,0,0,0,0,1,0,0,0,0,1],[9,37,0,0,5,80,0,0,0,0,88,0,0,0,0,88] >;

D4×Dic11 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_{11}
% in TeX

G:=Group("D4xDic11");
// GroupNames label

G:=SmallGroup(352,129);
// by ID

G=gap.SmallGroup(352,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,188,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^22=1,d^2=c^11,b*a*b=a^-1,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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