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## G = Q8.Dic11order 352 = 25·11

### The non-split extension by Q8 of Dic11 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — Q8.Dic11
 Chief series C1 — C11 — C22 — C44 — C11⋊C8 — C2×C11⋊C8 — Q8.Dic11
 Lower central C11 — C22 — Q8.Dic11
 Upper central C1 — C4 — C4○D4

Generators and relations for Q8.Dic11
G = < a,b,c,d | a4=b2=1, c22=a2, d2=a2c11, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c21 >

Subgroups: 178 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, D4, Q8, C11, C2×C8, M4(2), C4○D4, C22, C22, C8○D4, C44, C44, C2×C22, C11⋊C8, C11⋊C8, C2×C44, D4×C11, Q8×C11, C2×C11⋊C8, C44.C4, C11×C4○D4, Q8.Dic11
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D11, C8○D4, Dic11, D22, C2×Dic11, C22×D11, C22×Dic11, Q8.Dic11

Smallest permutation representation of Q8.Dic11
On 176 points
Generators in S176
(1 34 23 12)(2 35 24 13)(3 36 25 14)(4 37 26 15)(5 38 27 16)(6 39 28 17)(7 40 29 18)(8 41 30 19)(9 42 31 20)(10 43 32 21)(11 44 33 22)(45 56 67 78)(46 57 68 79)(47 58 69 80)(48 59 70 81)(49 60 71 82)(50 61 72 83)(51 62 73 84)(52 63 74 85)(53 64 75 86)(54 65 76 87)(55 66 77 88)(89 122 111 100)(90 123 112 101)(91 124 113 102)(92 125 114 103)(93 126 115 104)(94 127 116 105)(95 128 117 106)(96 129 118 107)(97 130 119 108)(98 131 120 109)(99 132 121 110)(133 144 155 166)(134 145 156 167)(135 146 157 168)(136 147 158 169)(137 148 159 170)(138 149 160 171)(139 150 161 172)(140 151 162 173)(141 152 163 174)(142 153 164 175)(143 154 165 176)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)(101 156)(102 157)(103 158)(104 159)(105 160)(106 161)(107 162)(108 163)(109 164)(110 165)(111 166)(112 167)(113 168)(114 169)(115 170)(116 171)(117 172)(118 173)(119 174)(120 175)(121 176)(122 133)(123 134)(124 135)(125 136)(126 137)(127 138)(128 139)(129 140)(130 141)(131 142)(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)
(1 122 34 111 23 100 12 89)(2 99 35 132 24 121 13 110)(3 120 36 109 25 98 14 131)(4 97 37 130 26 119 15 108)(5 118 38 107 27 96 16 129)(6 95 39 128 28 117 17 106)(7 116 40 105 29 94 18 127)(8 93 41 126 30 115 19 104)(9 114 42 103 31 92 20 125)(10 91 43 124 32 113 21 102)(11 112 44 101 33 90 22 123)(45 151 78 140 67 173 56 162)(46 172 79 161 68 150 57 139)(47 149 80 138 69 171 58 160)(48 170 81 159 70 148 59 137)(49 147 82 136 71 169 60 158)(50 168 83 157 72 146 61 135)(51 145 84 134 73 167 62 156)(52 166 85 155 74 144 63 133)(53 143 86 176 75 165 64 154)(54 164 87 153 76 142 65 175)(55 141 88 174 77 163 66 152)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E ··· 8J 11A ··· 11E 22A ··· 22E 22F ··· 22T 44A ··· 44J 44K ··· 44Y order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 8 ··· 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 size 1 1 2 2 2 1 1 2 2 2 11 11 11 11 22 ··· 22 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 D11 C8○D4 D22 Dic11 Dic11 Q8.Dic11 kernel Q8.Dic11 C2×C11⋊C8 C44.C4 C11×C4○D4 D4×C11 Q8×C11 C4○D4 C11 C2×C4 D4 Q8 C1 # reps 1 3 3 1 6 2 5 4 15 15 5 10

Matrix representation of Q8.Dic11 in GL4(𝔽89) generated by

 34 34 0 0 0 55 0 0 0 0 88 0 0 0 0 88
,
 34 34 0 0 21 55 0 0 0 0 1 0 0 0 0 1
,
 34 0 0 0 0 34 0 0 0 0 85 88 0 0 29 7
,
 52 0 0 0 0 52 0 0 0 0 71 47 0 0 48 18
G:=sub<GL(4,GF(89))| [34,0,0,0,34,55,0,0,0,0,88,0,0,0,0,88],[34,21,0,0,34,55,0,0,0,0,1,0,0,0,0,1],[34,0,0,0,0,34,0,0,0,0,85,29,0,0,88,7],[52,0,0,0,0,52,0,0,0,0,71,48,0,0,47,18] >;

Q8.Dic11 in GAP, Magma, Sage, TeX

Q_8.{\rm Dic}_{11}
% in TeX

G:=Group("Q8.Dic11");
// GroupNames label

G:=SmallGroup(352,143);
// by ID

G=gap.SmallGroup(352,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,188,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^22=a^2,d^2=a^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^21>;
// generators/relations

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