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## G = Dic6⋊5Q8order 192 = 26·3

### 3rd semidirect product of Dic6 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊5Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C4×Dic6 — Dic6⋊5Q8
 Lower central C3 — C6 — C2×C12 — Dic6⋊5Q8
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for Dic65Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a6b, dcd-1=c-1 >

Subgroups: 224 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C4 [×6], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×5], Dic3 [×3], C12 [×2], C12 [×2], C12 [×3], C2×C6, C42, C42, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×Q8 [×2], C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], Q8⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×Q8, C4⋊Q8, C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C6×Q8, C4.Q16, C12⋊C8, C6.Q16 [×2], C6.SD16 [×2], C4×Dic6, C3×C4⋊Q8, Dic65Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×Q16, C8⋊C22, C3⋊Q16 [×2], S3×Q8, Q83S3, C2×C3⋊D4, C4.Q16, D126C22, C2×C3⋊Q16, D63Q8, Dic65Q8

Smallest permutation representation of Dic65Q8
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)
(1 123 7 129)(2 122 8 128)(3 121 9 127)(4 132 10 126)(5 131 11 125)(6 130 12 124)(13 62 19 68)(14 61 20 67)(15 72 21 66)(16 71 22 65)(17 70 23 64)(18 69 24 63)(25 117 31 111)(26 116 32 110)(27 115 33 109)(28 114 34 120)(29 113 35 119)(30 112 36 118)(37 186 43 192)(38 185 44 191)(39 184 45 190)(40 183 46 189)(41 182 47 188)(42 181 48 187)(49 172 55 178)(50 171 56 177)(51 170 57 176)(52 169 58 175)(53 180 59 174)(54 179 60 173)(73 101 79 107)(74 100 80 106)(75 99 81 105)(76 98 82 104)(77 97 83 103)(78 108 84 102)(85 161 91 167)(86 160 92 166)(87 159 93 165)(88 158 94 164)(89 157 95 163)(90 168 96 162)(133 149 139 155)(134 148 140 154)(135 147 141 153)(136 146 142 152)(137 145 143 151)(138 156 144 150)
(1 164 100 109)(2 159 101 116)(3 166 102 111)(4 161 103 118)(5 168 104 113)(6 163 105 120)(7 158 106 115)(8 165 107 110)(9 160 108 117)(10 167 97 112)(11 162 98 119)(12 157 99 114)(13 172 154 192)(14 179 155 187)(15 174 156 182)(16 169 145 189)(17 176 146 184)(18 171 147 191)(19 178 148 186)(20 173 149 181)(21 180 150 188)(22 175 151 183)(23 170 152 190)(24 177 153 185)(25 124 86 81)(26 131 87 76)(27 126 88 83)(28 121 89 78)(29 128 90 73)(30 123 91 80)(31 130 92 75)(32 125 93 82)(33 132 94 77)(34 127 95 84)(35 122 96 79)(36 129 85 74)(37 65 55 137)(38 72 56 144)(39 67 57 139)(40 62 58 134)(41 69 59 141)(42 64 60 136)(43 71 49 143)(44 66 50 138)(45 61 51 133)(46 68 52 140)(47 63 53 135)(48 70 54 142)
(1 186 100 178)(2 187 101 179)(3 188 102 180)(4 189 103 169)(5 190 104 170)(6 191 105 171)(7 192 106 172)(8 181 107 173)(9 182 108 174)(10 183 97 175)(11 184 98 176)(12 185 99 177)(13 158 154 115)(14 159 155 116)(15 160 156 117)(16 161 145 118)(17 162 146 119)(18 163 147 120)(19 164 148 109)(20 165 149 110)(21 166 150 111)(22 167 151 112)(23 168 152 113)(24 157 153 114)(25 72 86 144)(26 61 87 133)(27 62 88 134)(28 63 89 135)(29 64 90 136)(30 65 91 137)(31 66 92 138)(32 67 93 139)(33 68 94 140)(34 69 95 141)(35 70 96 142)(36 71 85 143)(37 80 55 123)(38 81 56 124)(39 82 57 125)(40 83 58 126)(41 84 59 127)(42 73 60 128)(43 74 49 129)(44 75 50 130)(45 76 51 131)(46 77 52 132)(47 78 53 121)(48 79 54 122)

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

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

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

33 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 ··· 12F 12G 12H 12I 12J 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 size 1 1 1 1 2 2 2 2 2 4 8 8 12 12 12 12 2 2 2 12 12 12 12 4 ··· 4 8 8 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + - + - - + image C1 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 Q16 C4○D4 C3⋊D4 C8⋊C22 C3⋊Q16 S3×Q8 Q8⋊3S3 D12⋊6C22 kernel Dic6⋊5Q8 C12⋊C8 C6.Q16 C6.SD16 C4×Dic6 C3×C4⋊Q8 C4⋊Q8 Dic6 C2×C12 C42 C4⋊C4 C12 C12 C2×C4 C6 C4 C4 C4 C2 # reps 1 1 2 2 1 1 1 2 2 1 2 4 2 4 1 2 1 1 2

Matrix representation of Dic65Q8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 46 19 0 0 0 0 27 27 0 0 0 0 0 0 42 3 0 0 0 0 45 31 0 0 0 0 0 0 19 21 0 0 0 0 21 54
,
 27 0 0 0 0 0 46 46 0 0 0 0 0 0 43 60 0 0 0 0 13 30 0 0 0 0 0 0 32 17 0 0 0 0 17 41
,
 1 2 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[46,27,0,0,0,0,19,27,0,0,0,0,0,0,42,45,0,0,0,0,3,31,0,0,0,0,0,0,19,21,0,0,0,0,21,54],[27,46,0,0,0,0,0,46,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,0,0,0,0,32,17,0,0,0,0,17,41],[1,72,0,0,0,0,2,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0] >;

Dic65Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_5Q_8
% in TeX

G:=Group("Dic6:5Q8");
// GroupNames label

G:=SmallGroup(192,650);
// by ID

G=gap.SmallGroup(192,650);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,268,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^7,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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