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## G = Q8×C3⋊C8order 192 = 26·3

### Direct product of Q8 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×C3⋊C8
G = < a,b,c,d | a4=c3=d8=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 152 in 102 conjugacy classes, 77 normal (22 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C3×Q8, C4×C8, C4⋊C8, C4×Q8, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C3×C4⋊C4, C6×Q8, C8×Q8, C4×C3⋊C8, C12⋊C8, Q8×C12, Q8×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Q8, C23, Dic3, D6, C2×C8, C22×C4, C2×Q8, C4○D4, C3⋊C8, C2×Dic3, C22×S3, C4×Q8, C22×C8, C8○D4, C2×C3⋊C8, S3×Q8, Q83S3, C22×Dic3, C8×Q8, C22×C3⋊C8, Q8×Dic3, D4.Dic3, Q8×C3⋊C8

Smallest permutation representation of Q8×C3⋊C8
Regular action on 192 points
Generators in S192
(1 51 131 171)(2 52 132 172)(3 53 133 173)(4 54 134 174)(5 55 135 175)(6 56 136 176)(7 49 129 169)(8 50 130 170)(9 65 151 89)(10 66 152 90)(11 67 145 91)(12 68 146 92)(13 69 147 93)(14 70 148 94)(15 71 149 95)(16 72 150 96)(17 29 178 98)(18 30 179 99)(19 31 180 100)(20 32 181 101)(21 25 182 102)(22 26 183 103)(23 27 184 104)(24 28 177 97)(33 117 85 161)(34 118 86 162)(35 119 87 163)(36 120 88 164)(37 113 81 165)(38 114 82 166)(39 115 83 167)(40 116 84 168)(41 77 157 123)(42 78 158 124)(43 79 159 125)(44 80 160 126)(45 73 153 127)(46 74 154 128)(47 75 155 121)(48 76 156 122)(57 185 105 141)(58 186 106 142)(59 187 107 143)(60 188 108 144)(61 189 109 137)(62 190 110 138)(63 191 111 139)(64 192 112 140)
(1 67 131 91)(2 68 132 92)(3 69 133 93)(4 70 134 94)(5 71 135 95)(6 72 136 96)(7 65 129 89)(8 66 130 90)(9 169 151 49)(10 170 152 50)(11 171 145 51)(12 172 146 52)(13 173 147 53)(14 174 148 54)(15 175 149 55)(16 176 150 56)(17 42 178 158)(18 43 179 159)(19 44 180 160)(20 45 181 153)(21 46 182 154)(22 47 183 155)(23 48 184 156)(24 41 177 157)(25 128 102 74)(26 121 103 75)(27 122 104 76)(28 123 97 77)(29 124 98 78)(30 125 99 79)(31 126 100 80)(32 127 101 73)(33 109 85 61)(34 110 86 62)(35 111 87 63)(36 112 88 64)(37 105 81 57)(38 106 82 58)(39 107 83 59)(40 108 84 60)(113 185 165 141)(114 186 166 142)(115 187 167 143)(116 188 168 144)(117 189 161 137)(118 190 162 138)(119 191 163 139)(120 192 164 140)
(1 41 115)(2 116 42)(3 43 117)(4 118 44)(5 45 119)(6 120 46)(7 47 113)(8 114 48)(9 26 57)(10 58 27)(11 28 59)(12 60 29)(13 30 61)(14 62 31)(15 32 63)(16 64 25)(17 92 144)(18 137 93)(19 94 138)(20 139 95)(21 96 140)(22 141 89)(23 90 142)(24 143 91)(33 173 125)(34 126 174)(35 175 127)(36 128 176)(37 169 121)(38 122 170)(39 171 123)(40 124 172)(49 75 81)(50 82 76)(51 77 83)(52 84 78)(53 79 85)(54 86 80)(55 73 87)(56 88 74)(65 183 185)(66 186 184)(67 177 187)(68 188 178)(69 179 189)(70 190 180)(71 181 191)(72 192 182)(97 107 145)(98 146 108)(99 109 147)(100 148 110)(101 111 149)(102 150 112)(103 105 151)(104 152 106)(129 155 165)(130 166 156)(131 157 167)(132 168 158)(133 159 161)(134 162 160)(135 153 163)(136 164 154)
(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)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E ··· 4P 6A 6B 6C 8A ··· 8H 8I ··· 8T 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 3 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 1 1 1 1 2 ··· 2 2 2 2 3 ··· 3 6 ··· 6 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + - + - - - + image C1 C2 C2 C2 C4 C4 C8 S3 Q8 D6 Dic3 Dic3 C4○D4 C3⋊C8 C8○D4 S3×Q8 Q8⋊3S3 D4.Dic3 kernel Q8×C3⋊C8 C4×C3⋊C8 C12⋊C8 Q8×C12 C3×C4⋊C4 C6×Q8 C3×Q8 C4×Q8 C3⋊C8 C42 C4⋊C4 C2×Q8 C12 Q8 C6 C4 C4 C2 # reps 1 3 3 1 6 2 16 1 2 3 3 1 2 8 4 1 1 2

Matrix representation of Q8×C3⋊C8 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 0 1 0 0 72 0
,
 72 0 0 0 0 72 0 0 0 0 62 30 0 0 30 11
,
 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 23 62 0 0 39 50 0 0 0 0 46 0 0 0 0 46
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,62,30,0,0,30,11],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[23,39,0,0,62,50,0,0,0,0,46,0,0,0,0,46] >;

Q8×C3⋊C8 in GAP, Magma, Sage, TeX

Q_8\times C_3\rtimes C_8
% in TeX

G:=Group("Q8xC3:C8");
// GroupNames label

G:=SmallGroup(192,582);
// by ID

G=gap.SmallGroup(192,582);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,219,100,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^3=d^8=1,b^2=a^2,b*a*b^-1=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

׿
×
𝔽