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G = Dic6⋊3Q8order 192 = 26·3

1st semidirect product of Dic6 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊3Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C4×Dic6 — Dic6⋊3Q8
 Lower central C3 — C6 — C2×C12 — Dic6⋊3Q8
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for Dic63Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=cac-1=a-1, ad=da, cbc-1=a9b, dbd-1=a6b, dcd-1=c-1 >

Subgroups: 264 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×Dic6, C4.Q16, C2.Dic12, C241C4, C3×C4⋊C8, C4×Dic6, C122Q8, Dic63Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, Q16, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C2×Q16, C8⋊C22, Dic12, C2×D12, D42S3, S3×Q8, C4.Q16, C4.D12, C2×Dic12, C8⋊D6, Dic63Q8

Smallest permutation representation of Dic63Q8
Regular action on 192 points
Generators in S192
(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)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 187 7 181)(2 186 8 192)(3 185 9 191)(4 184 10 190)(5 183 11 189)(6 182 12 188)(13 126 19 132)(14 125 20 131)(15 124 21 130)(16 123 22 129)(17 122 23 128)(18 121 24 127)(25 91 31 85)(26 90 32 96)(27 89 33 95)(28 88 34 94)(29 87 35 93)(30 86 36 92)(37 57 43 51)(38 56 44 50)(39 55 45 49)(40 54 46 60)(41 53 47 59)(42 52 48 58)(61 166 67 160)(62 165 68 159)(63 164 69 158)(64 163 70 157)(65 162 71 168)(66 161 72 167)(73 177 79 171)(74 176 80 170)(75 175 81 169)(76 174 82 180)(77 173 83 179)(78 172 84 178)(97 115 103 109)(98 114 104 120)(99 113 105 119)(100 112 106 118)(101 111 107 117)(102 110 108 116)(133 145 139 151)(134 156 140 150)(135 155 141 149)(136 154 142 148)(137 153 143 147)(138 152 144 146)
(1 30 105 179)(2 29 106 178)(3 28 107 177)(4 27 108 176)(5 26 97 175)(6 25 98 174)(7 36 99 173)(8 35 100 172)(9 34 101 171)(10 33 102 170)(11 32 103 169)(12 31 104 180)(13 38 168 145)(14 37 157 156)(15 48 158 155)(16 47 159 154)(17 46 160 153)(18 45 161 152)(19 44 162 151)(20 43 163 150)(21 42 164 149)(22 41 165 148)(23 40 166 147)(24 39 167 146)(49 63 144 124)(50 62 133 123)(51 61 134 122)(52 72 135 121)(53 71 136 132)(54 70 137 131)(55 69 138 130)(56 68 139 129)(57 67 140 128)(58 66 141 127)(59 65 142 126)(60 64 143 125)(73 182 94 114)(74 181 95 113)(75 192 96 112)(76 191 85 111)(77 190 86 110)(78 189 87 109)(79 188 88 120)(80 187 89 119)(81 186 90 118)(82 185 91 117)(83 184 92 116)(84 183 93 115)
(1 135 105 52)(2 136 106 53)(3 137 107 54)(4 138 108 55)(5 139 97 56)(6 140 98 57)(7 141 99 58)(8 142 100 59)(9 143 101 60)(10 144 102 49)(11 133 103 50)(12 134 104 51)(13 93 168 84)(14 94 157 73)(15 95 158 74)(16 96 159 75)(17 85 160 76)(18 86 161 77)(19 87 162 78)(20 88 163 79)(21 89 164 80)(22 90 165 81)(23 91 166 82)(24 92 167 83)(25 67 174 128)(26 68 175 129)(27 69 176 130)(28 70 177 131)(29 71 178 132)(30 72 179 121)(31 61 180 122)(32 62 169 123)(33 63 170 124)(34 64 171 125)(35 65 172 126)(36 66 173 127)(37 182 156 114)(38 183 145 115)(39 184 146 116)(40 185 147 117)(41 186 148 118)(42 187 149 119)(43 188 150 120)(44 189 151 109)(45 190 152 110)(46 191 153 111)(47 192 154 112)(48 181 155 113)

G:=sub<Sym(192)| (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)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,187,7,181)(2,186,8,192)(3,185,9,191)(4,184,10,190)(5,183,11,189)(6,182,12,188)(13,126,19,132)(14,125,20,131)(15,124,21,130)(16,123,22,129)(17,122,23,128)(18,121,24,127)(25,91,31,85)(26,90,32,96)(27,89,33,95)(28,88,34,94)(29,87,35,93)(30,86,36,92)(37,57,43,51)(38,56,44,50)(39,55,45,49)(40,54,46,60)(41,53,47,59)(42,52,48,58)(61,166,67,160)(62,165,68,159)(63,164,69,158)(64,163,70,157)(65,162,71,168)(66,161,72,167)(73,177,79,171)(74,176,80,170)(75,175,81,169)(76,174,82,180)(77,173,83,179)(78,172,84,178)(97,115,103,109)(98,114,104,120)(99,113,105,119)(100,112,106,118)(101,111,107,117)(102,110,108,116)(133,145,139,151)(134,156,140,150)(135,155,141,149)(136,154,142,148)(137,153,143,147)(138,152,144,146), (1,30,105,179)(2,29,106,178)(3,28,107,177)(4,27,108,176)(5,26,97,175)(6,25,98,174)(7,36,99,173)(8,35,100,172)(9,34,101,171)(10,33,102,170)(11,32,103,169)(12,31,104,180)(13,38,168,145)(14,37,157,156)(15,48,158,155)(16,47,159,154)(17,46,160,153)(18,45,161,152)(19,44,162,151)(20,43,163,150)(21,42,164,149)(22,41,165,148)(23,40,166,147)(24,39,167,146)(49,63,144,124)(50,62,133,123)(51,61,134,122)(52,72,135,121)(53,71,136,132)(54,70,137,131)(55,69,138,130)(56,68,139,129)(57,67,140,128)(58,66,141,127)(59,65,142,126)(60,64,143,125)(73,182,94,114)(74,181,95,113)(75,192,96,112)(76,191,85,111)(77,190,86,110)(78,189,87,109)(79,188,88,120)(80,187,89,119)(81,186,90,118)(82,185,91,117)(83,184,92,116)(84,183,93,115), (1,135,105,52)(2,136,106,53)(3,137,107,54)(4,138,108,55)(5,139,97,56)(6,140,98,57)(7,141,99,58)(8,142,100,59)(9,143,101,60)(10,144,102,49)(11,133,103,50)(12,134,104,51)(13,93,168,84)(14,94,157,73)(15,95,158,74)(16,96,159,75)(17,85,160,76)(18,86,161,77)(19,87,162,78)(20,88,163,79)(21,89,164,80)(22,90,165,81)(23,91,166,82)(24,92,167,83)(25,67,174,128)(26,68,175,129)(27,69,176,130)(28,70,177,131)(29,71,178,132)(30,72,179,121)(31,61,180,122)(32,62,169,123)(33,63,170,124)(34,64,171,125)(35,65,172,126)(36,66,173,127)(37,182,156,114)(38,183,145,115)(39,184,146,116)(40,185,147,117)(41,186,148,118)(42,187,149,119)(43,188,150,120)(44,189,151,109)(45,190,152,110)(46,191,153,111)(47,192,154,112)(48,181,155,113)>;

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)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,187,7,181)(2,186,8,192)(3,185,9,191)(4,184,10,190)(5,183,11,189)(6,182,12,188)(13,126,19,132)(14,125,20,131)(15,124,21,130)(16,123,22,129)(17,122,23,128)(18,121,24,127)(25,91,31,85)(26,90,32,96)(27,89,33,95)(28,88,34,94)(29,87,35,93)(30,86,36,92)(37,57,43,51)(38,56,44,50)(39,55,45,49)(40,54,46,60)(41,53,47,59)(42,52,48,58)(61,166,67,160)(62,165,68,159)(63,164,69,158)(64,163,70,157)(65,162,71,168)(66,161,72,167)(73,177,79,171)(74,176,80,170)(75,175,81,169)(76,174,82,180)(77,173,83,179)(78,172,84,178)(97,115,103,109)(98,114,104,120)(99,113,105,119)(100,112,106,118)(101,111,107,117)(102,110,108,116)(133,145,139,151)(134,156,140,150)(135,155,141,149)(136,154,142,148)(137,153,143,147)(138,152,144,146), (1,30,105,179)(2,29,106,178)(3,28,107,177)(4,27,108,176)(5,26,97,175)(6,25,98,174)(7,36,99,173)(8,35,100,172)(9,34,101,171)(10,33,102,170)(11,32,103,169)(12,31,104,180)(13,38,168,145)(14,37,157,156)(15,48,158,155)(16,47,159,154)(17,46,160,153)(18,45,161,152)(19,44,162,151)(20,43,163,150)(21,42,164,149)(22,41,165,148)(23,40,166,147)(24,39,167,146)(49,63,144,124)(50,62,133,123)(51,61,134,122)(52,72,135,121)(53,71,136,132)(54,70,137,131)(55,69,138,130)(56,68,139,129)(57,67,140,128)(58,66,141,127)(59,65,142,126)(60,64,143,125)(73,182,94,114)(74,181,95,113)(75,192,96,112)(76,191,85,111)(77,190,86,110)(78,189,87,109)(79,188,88,120)(80,187,89,119)(81,186,90,118)(82,185,91,117)(83,184,92,116)(84,183,93,115), (1,135,105,52)(2,136,106,53)(3,137,107,54)(4,138,108,55)(5,139,97,56)(6,140,98,57)(7,141,99,58)(8,142,100,59)(9,143,101,60)(10,144,102,49)(11,133,103,50)(12,134,104,51)(13,93,168,84)(14,94,157,73)(15,95,158,74)(16,96,159,75)(17,85,160,76)(18,86,161,77)(19,87,162,78)(20,88,163,79)(21,89,164,80)(22,90,165,81)(23,91,166,82)(24,92,167,83)(25,67,174,128)(26,68,175,129)(27,69,176,130)(28,70,177,131)(29,71,178,132)(30,72,179,121)(31,61,180,122)(32,62,169,123)(33,63,170,124)(34,64,171,125)(35,65,172,126)(36,66,173,127)(37,182,156,114)(38,183,145,115)(39,184,146,116)(40,185,147,117)(41,186,148,118)(42,187,149,119)(43,188,150,120)(44,189,151,109)(45,190,152,110)(46,191,153,111)(47,192,154,112)(48,181,155,113) );

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),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,187,7,181),(2,186,8,192),(3,185,9,191),(4,184,10,190),(5,183,11,189),(6,182,12,188),(13,126,19,132),(14,125,20,131),(15,124,21,130),(16,123,22,129),(17,122,23,128),(18,121,24,127),(25,91,31,85),(26,90,32,96),(27,89,33,95),(28,88,34,94),(29,87,35,93),(30,86,36,92),(37,57,43,51),(38,56,44,50),(39,55,45,49),(40,54,46,60),(41,53,47,59),(42,52,48,58),(61,166,67,160),(62,165,68,159),(63,164,69,158),(64,163,70,157),(65,162,71,168),(66,161,72,167),(73,177,79,171),(74,176,80,170),(75,175,81,169),(76,174,82,180),(77,173,83,179),(78,172,84,178),(97,115,103,109),(98,114,104,120),(99,113,105,119),(100,112,106,118),(101,111,107,117),(102,110,108,116),(133,145,139,151),(134,156,140,150),(135,155,141,149),(136,154,142,148),(137,153,143,147),(138,152,144,146)], [(1,30,105,179),(2,29,106,178),(3,28,107,177),(4,27,108,176),(5,26,97,175),(6,25,98,174),(7,36,99,173),(8,35,100,172),(9,34,101,171),(10,33,102,170),(11,32,103,169),(12,31,104,180),(13,38,168,145),(14,37,157,156),(15,48,158,155),(16,47,159,154),(17,46,160,153),(18,45,161,152),(19,44,162,151),(20,43,163,150),(21,42,164,149),(22,41,165,148),(23,40,166,147),(24,39,167,146),(49,63,144,124),(50,62,133,123),(51,61,134,122),(52,72,135,121),(53,71,136,132),(54,70,137,131),(55,69,138,130),(56,68,139,129),(57,67,140,128),(58,66,141,127),(59,65,142,126),(60,64,143,125),(73,182,94,114),(74,181,95,113),(75,192,96,112),(76,191,85,111),(77,190,86,110),(78,189,87,109),(79,188,88,120),(80,187,89,119),(81,186,90,118),(82,185,91,117),(83,184,92,116),(84,183,93,115)], [(1,135,105,52),(2,136,106,53),(3,137,107,54),(4,138,108,55),(5,139,97,56),(6,140,98,57),(7,141,99,58),(8,142,100,59),(9,143,101,60),(10,144,102,49),(11,133,103,50),(12,134,104,51),(13,93,168,84),(14,94,157,73),(15,95,158,74),(16,96,159,75),(17,85,160,76),(18,86,161,77),(19,87,162,78),(20,88,163,79),(21,89,164,80),(22,90,165,81),(23,91,166,82),(24,92,167,83),(25,67,174,128),(26,68,175,129),(27,69,176,130),(28,70,177,131),(29,71,178,132),(30,72,179,121),(31,61,180,122),(32,62,169,123),(33,63,170,124),(34,64,171,125),(35,65,172,126),(36,66,173,127),(37,182,156,114),(38,183,145,115),(39,184,146,116),(40,185,147,117),(41,186,148,118),(42,187,149,119),(43,188,150,120),(44,189,151,109),(45,190,152,110),(46,191,153,111),(47,192,154,112),(48,181,155,113)]])

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 2 2 4 12 12 12 12 24 24 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + - + - + - - + image C1 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 Q16 C4○D4 D12 Dic12 C8⋊C22 D4⋊2S3 S3×Q8 C8⋊D6 kernel Dic6⋊3Q8 C2.Dic12 C24⋊1C4 C3×C4⋊C8 C4×Dic6 C12⋊2Q8 C4⋊C8 Dic6 C2×C12 C42 C2×C8 C12 C12 C2×C4 C4 C6 C4 C4 C2 # reps 1 2 2 1 1 1 1 2 2 1 2 4 2 4 8 1 1 1 2

Matrix representation of Dic63Q8 in GL4(𝔽73) generated by

 59 66 0 0 7 66 0 0 0 0 72 0 0 0 0 72
,
 27 0 0 0 46 46 0 0 0 0 67 10 0 0 33 6
,
 48 11 0 0 36 25 0 0 0 0 4 6 0 0 58 69
,
 7 14 0 0 59 66 0 0 0 0 16 22 0 0 58 57
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,72,0,0,0,0,72],[27,46,0,0,0,46,0,0,0,0,67,33,0,0,10,6],[48,36,0,0,11,25,0,0,0,0,4,58,0,0,6,69],[7,59,0,0,14,66,0,0,0,0,16,58,0,0,22,57] >;

Dic63Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_3Q_8
% in TeX

G:=Group("Dic6:3Q8");
// GroupNames label

G:=SmallGroup(192,409);
// by ID

G=gap.SmallGroup(192,409);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,310,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations

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