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G = C12⋊Q16order 192 = 26·3

2nd semidirect product of C12 and Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C122Q16, C42.83D6, Dic6.24D4, C4⋊Q8.9S3, C4.57(S3×D4), C42(C3⋊Q16), C12.39(C2×D4), C34(C42Q16), (C2×Q8).71D6, C6.42(C2×Q16), (C2×C12).159D4, C12⋊C8.21C2, C12.85(C4○D4), C4.6(D42S3), (C4×Dic6).17C2, (C6×Q8).65C22, C2.15(D63D4), C6.106(C4⋊D4), (C2×C12).408C23, (C4×C12).137C22, Q82Dic3.13C2, C6.99(C8.C22), C4⋊Dic3.349C22, C2.20(Q8.11D6), (C2×Dic6).275C22, (C3×C4⋊Q8).9C2, (C2×C6).539(C2×D4), (C2×C3⋊Q16).6C2, C2.13(C2×C3⋊Q16), (C2×C3⋊C8).140C22, (C2×C4).190(C3⋊D4), (C2×C4).505(C22×S3), C22.211(C2×C3⋊D4), SmallGroup(192,649)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C12⋊Q16
C1C3C6C12C2×C12C2×Dic6C4×Dic6 — C12⋊Q16
C3C6C2×C12 — C12⋊Q16
C1C22C42C4⋊Q8

Generators and relations for C12⋊Q16
 G = < a,b,c | a12=b8=1, c2=b4, bab-1=a-1, cac-1=a7, cbc-1=b-1 >

Subgroups: 256 in 108 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 [×7], Dic3 [×3], C12 [×2], C12 [×2], C12 [×3], C2×C6, C42, C42, C4⋊C4 [×4], C2×C8 [×2], Q16 [×4], C2×Q8 [×2], C2×Q8, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×4], Q8⋊C4 [×2], C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16 [×2], C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×4], C4×C12, C3×C4⋊C4 [×2], C2×Dic6, C6×Q8 [×2], C42Q16, C12⋊C8, Q82Dic3 [×2], C4×Dic6, C2×C3⋊Q16 [×2], C3×C4⋊Q8, C12⋊Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], Q16 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×Q16, C8.C22, C3⋊Q16 [×2], S3×D4, D42S3, C2×C3⋊D4, C42Q16, D63D4, Q8.11D6, C2×C3⋊Q16, C12⋊Q16

Smallest permutation representation of C12⋊Q16
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)
(1 159 83 146 69 121 29 44)(2 158 84 145 70 132 30 43)(3 157 73 156 71 131 31 42)(4 168 74 155 72 130 32 41)(5 167 75 154 61 129 33 40)(6 166 76 153 62 128 34 39)(7 165 77 152 63 127 35 38)(8 164 78 151 64 126 36 37)(9 163 79 150 65 125 25 48)(10 162 80 149 66 124 26 47)(11 161 81 148 67 123 27 46)(12 160 82 147 68 122 28 45)(13 108 88 50 176 139 120 192)(14 107 89 49 177 138 109 191)(15 106 90 60 178 137 110 190)(16 105 91 59 179 136 111 189)(17 104 92 58 180 135 112 188)(18 103 93 57 169 134 113 187)(19 102 94 56 170 133 114 186)(20 101 95 55 171 144 115 185)(21 100 96 54 172 143 116 184)(22 99 85 53 173 142 117 183)(23 98 86 52 174 141 118 182)(24 97 87 51 175 140 119 181)
(1 106 69 137)(2 101 70 144)(3 108 71 139)(4 103 72 134)(5 98 61 141)(6 105 62 136)(7 100 63 143)(8 107 64 138)(9 102 65 133)(10 97 66 140)(11 104 67 135)(12 99 68 142)(13 131 176 157)(14 126 177 164)(15 121 178 159)(16 128 179 166)(17 123 180 161)(18 130 169 168)(19 125 170 163)(20 132 171 158)(21 127 172 165)(22 122 173 160)(23 129 174 167)(24 124 175 162)(25 56 79 186)(26 51 80 181)(27 58 81 188)(28 53 82 183)(29 60 83 190)(30 55 84 185)(31 50 73 192)(32 57 74 187)(33 52 75 182)(34 59 76 189)(35 54 77 184)(36 49 78 191)(37 89 151 109)(38 96 152 116)(39 91 153 111)(40 86 154 118)(41 93 155 113)(42 88 156 120)(43 95 145 115)(44 90 146 110)(45 85 147 117)(46 92 148 112)(47 87 149 119)(48 94 150 114)

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

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

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

33 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K6A6B6C8A8B8C8D12A···12F12G12H12I12J
order1222344444444444666888812···1212121212
size11112222248812121212222121212124···48888

33 irreducible representations

dim1111112222222244444
type+++++++++++---+-
imageC1C2C2C2C2C2S3D4D4D6D6Q16C4○D4C3⋊D4C8.C22C3⋊Q16S3×D4D42S3Q8.11D6
kernelC12⋊Q16C12⋊C8Q82Dic3C4×Dic6C2×C3⋊Q16C3×C4⋊Q8C4⋊Q8Dic6C2×C12C42C2×Q8C12C12C2×C4C6C4C4C4C2
# reps1121211221242412112

Matrix representation of C12⋊Q16 in GL6(𝔽73)

7200000
0720000
00415200
00213200
0000172
000010
,
16570000
16160000
000100
001000
0000721
000001
,
660000
6670000
0007200
0072000
000010
000001

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,41,21,0,0,0,0,52,32,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C12⋊Q16 in GAP, Magma, Sage, TeX

C_{12}\rtimes Q_{16}
% in TeX

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

G:=SmallGroup(192,649);
// by ID

G=gap.SmallGroup(192,649);
# by ID

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

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

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