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## G = C12⋊3Q16order 192 = 26·3

### 3rd semidirect product of C12 and Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12⋊3Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12⋊2Q8 — C12⋊3Q16
 Lower central C3 — C6 — C2×C12 — C12⋊3Q16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C123Q16
G = < a,b,c | a12=b8=1, c2=b4, bab-1=a5, cac-1=a7, cbc-1=b-1 >

Subgroups: 304 in 122 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C4 [×4], C22, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×8], Dic3 [×2], C12 [×6], C12 [×2], C2×C6, C42, C4⋊C4 [×4], C2×C8 [×2], Q16 [×8], C2×Q8 [×2], C2×Q8 [×2], C3⋊C8 [×4], Dic6 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C2×C12 [×2], C3×Q8 [×4], C4×C8, C4⋊Q8, C4⋊Q8, C2×Q16 [×4], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C3⋊Q16 [×8], C4×C12, C3×C4⋊C4 [×2], C2×Dic6 [×2], C6×Q8 [×2], C4⋊Q16, C4×C3⋊C8, C122Q8, C2×C3⋊Q16 [×4], C3×C4⋊Q8, C123Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], Q16 [×4], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C2×Q16 [×2], C3⋊Q16 [×4], S3×D4 [×2], C2×C3⋊D4, C4⋊Q16, C123D4, C2×C3⋊Q16 [×2], C123Q16

Smallest permutation representation of C123Q16
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)
(1 118 99 69 41 133 121 166)(2 111 100 62 42 138 122 159)(3 116 101 67 43 143 123 164)(4 109 102 72 44 136 124 157)(5 114 103 65 45 141 125 162)(6 119 104 70 46 134 126 167)(7 112 105 63 47 139 127 160)(8 117 106 68 48 144 128 165)(9 110 107 61 37 137 129 158)(10 115 108 66 38 142 130 163)(11 120 97 71 39 135 131 168)(12 113 98 64 40 140 132 161)(13 151 172 80 58 187 86 29)(14 156 173 73 59 192 87 34)(15 149 174 78 60 185 88 27)(16 154 175 83 49 190 89 32)(17 147 176 76 50 183 90 25)(18 152 177 81 51 188 91 30)(19 145 178 74 52 181 92 35)(20 150 179 79 53 186 93 28)(21 155 180 84 54 191 94 33)(22 148 169 77 55 184 95 26)(23 153 170 82 56 189 96 31)(24 146 171 75 57 182 85 36)
(1 52 41 19)(2 59 42 14)(3 54 43 21)(4 49 44 16)(5 56 45 23)(6 51 46 18)(7 58 47 13)(8 53 48 20)(9 60 37 15)(10 55 38 22)(11 50 39 17)(12 57 40 24)(25 120 76 135)(26 115 77 142)(27 110 78 137)(28 117 79 144)(29 112 80 139)(30 119 81 134)(31 114 82 141)(32 109 83 136)(33 116 84 143)(34 111 73 138)(35 118 74 133)(36 113 75 140)(61 149 158 185)(62 156 159 192)(63 151 160 187)(64 146 161 182)(65 153 162 189)(66 148 163 184)(67 155 164 191)(68 150 165 186)(69 145 166 181)(70 152 167 188)(71 147 168 183)(72 154 157 190)(85 98 171 132)(86 105 172 127)(87 100 173 122)(88 107 174 129)(89 102 175 124)(90 97 176 131)(91 104 177 126)(92 99 178 121)(93 106 179 128)(94 101 180 123)(95 108 169 130)(96 103 170 125)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - - + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 Q16 C3⋊D4 C3⋊Q16 S3×D4 kernel C12⋊3Q16 C4×C3⋊C8 C12⋊2Q8 C2×C3⋊Q16 C3×C4⋊Q8 C4⋊Q8 C3⋊C8 C2×C12 C42 C2×Q8 C12 C2×C4 C4 C4 # reps 1 1 1 4 1 1 4 2 1 2 8 4 4 2

Matrix representation of C123Q16 in GL6(𝔽73)

 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 56 12 0 0 0 0 61 17
,
 16 57 0 0 0 0 16 16 0 0 0 0 0 0 22 32 0 0 0 0 10 51 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 30 62 0 0 0 0 62 43 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,56,61,0,0,0,0,12,17],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,22,10,0,0,0,0,32,51,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[30,62,0,0,0,0,62,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C123Q16 in GAP, Magma, Sage, TeX

C_{12}\rtimes_3Q_{16}
% in TeX

G:=Group("C12:3Q16");
// GroupNames label

G:=SmallGroup(192,651);
// by ID

G=gap.SmallGroup(192,651);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,219,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations

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