Copied to
clipboard

## G = Dic11.D4order 352 = 25·11

### 1st non-split extension by Dic11 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — Dic11.D4
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — D22⋊C4 — Dic11.D4
 Lower central C11 — C2×C22 — Dic11.D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic11.D4
G = < a,b,c,d | a22=c4=1, b2=d2=a11, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a11b, dcd-1=a11c-1 >

Subgroups: 466 in 76 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C11, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, D11, C22, C22, C4.4D4, Dic11, Dic11, C44, D22, C2×C22, C2×C22, Dic22, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×C22, C4×Dic11, D22⋊C4, C23.D11, C11×C22⋊C4, C2×Dic22, C2×C11⋊D4, Dic11.D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4.4D4, D22, C22×D11, D445C2, D4×D11, D42D11, Dic11.D4

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

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 11A ··· 11E 22A ··· 22O 22P ··· 22Y 44A ··· 44T order 1 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 4 44 2 2 4 22 22 22 22 44 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

64 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 C2 D4 C4○D4 D11 D22 D22 D44⋊5C2 D4×D11 D4⋊2D11 kernel Dic11.D4 C4×Dic11 D22⋊C4 C23.D11 C11×C22⋊C4 C2×Dic22 C2×C11⋊D4 Dic11 C22 C22⋊C4 C2×C4 C23 C2 C2 C2 # reps 1 1 2 1 1 1 1 2 4 5 10 5 20 5 5

Matrix representation of Dic11.D4 in GL6(𝔽89)

 51 1 0 0 0 0 62 85 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 71 51 0 0 0 0 53 18 0 0 0 0 0 0 1 47 0 0 0 0 17 88 0 0 0 0 0 0 88 0 0 0 0 0 0 88
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 55 0 0 0 0 0 0 55 0 0 0 0 0 0 19 87 0 0 0 0 3 70
,
 18 38 0 0 0 0 36 71 0 0 0 0 0 0 34 85 0 0 0 0 0 55 0 0 0 0 0 0 70 2 0 0 0 0 87 19

G:=sub<GL(6,GF(89))| [51,62,0,0,0,0,1,85,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[71,53,0,0,0,0,51,18,0,0,0,0,0,0,1,17,0,0,0,0,47,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,19,3,0,0,0,0,87,70],[18,36,0,0,0,0,38,71,0,0,0,0,0,0,34,0,0,0,0,0,85,55,0,0,0,0,0,0,70,87,0,0,0,0,2,19] >;

Dic11.D4 in GAP, Magma, Sage, TeX

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

G:=Group("Dic11.D4");
// GroupNames label

G:=SmallGroup(352,80);
// by ID

G=gap.SmallGroup(352,80);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,55,506,188,11525]);
// Polycyclic

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

׿
×
𝔽