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G = C127Q16order 192 = 26·3

1st semidirect product of C12 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C127Q16, Q8.7D12, C42.60D6, C4⋊C4.257D6, C43(C3⋊Q16), C12.22(C2×D4), (C2×C12).68D4, C4.18(C2×D12), C33(C42Q16), (Q8×C12).9C2, (C3×Q8).19D4, (C4×Q8).15S3, C6.36(C2×Q16), (C2×Q8).188D6, C12⋊C8.19C2, C12.64(C4○D4), C4.14(C4○D12), C6.69(C4⋊D4), C122Q8.15C2, C2.17(C127D4), (C2×C12).351C23, (C4×C12).102C22, C6.SD16.11C2, (C6×Q8).199C22, C2.10(Q8.14D6), C6.112(C8.C22), (C2×Dic6).102C22, C2.7(C2×C3⋊Q16), (C2×C6).482(C2×D4), (C2×C3⋊Q16).5C2, (C2×C3⋊C8).104C22, (C2×C4).250(C3⋊D4), (C3×C4⋊C4).288C22, (C2×C4).451(C22×S3), C22.157(C2×C3⋊D4), SmallGroup(192,590)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C127Q16
C1C3C6C12C2×C12C2×Dic6C122Q8 — C127Q16
C3C6C2×C12 — C127Q16
C1C22C42C4×Q8

Generators and relations for C127Q16
 G = < a,b,c | a12=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 264 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C4 [×6], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×5], Dic3 [×2], C12 [×2], C12 [×2], C12 [×4], C2×C6, C42, C42, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8 [×2], C3⋊C8 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, Q8⋊C4 [×2], C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16 [×2], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C3⋊Q16 [×4], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6 [×2], C6×Q8, C42Q16, C12⋊C8, C6.SD16 [×2], C122Q8, C2×C3⋊Q16 [×2], Q8×C12, C127Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], Q16 [×2], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×Q16, C8.C22, C3⋊Q16 [×2], C2×D12, C4○D12, C2×C3⋊D4, C42Q16, C127D4, C2×C3⋊Q16, Q8.14D6, C127Q16

Smallest permutation representation of C127Q16
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 93 130 171 68 164 136 97)(2 92 131 170 69 163 137 108)(3 91 132 169 70 162 138 107)(4 90 121 180 71 161 139 106)(5 89 122 179 72 160 140 105)(6 88 123 178 61 159 141 104)(7 87 124 177 62 158 142 103)(8 86 125 176 63 157 143 102)(9 85 126 175 64 168 144 101)(10 96 127 174 65 167 133 100)(11 95 128 173 66 166 134 99)(12 94 129 172 67 165 135 98)(13 187 112 56 29 38 78 156)(14 186 113 55 30 37 79 155)(15 185 114 54 31 48 80 154)(16 184 115 53 32 47 81 153)(17 183 116 52 33 46 82 152)(18 182 117 51 34 45 83 151)(19 181 118 50 35 44 84 150)(20 192 119 49 36 43 73 149)(21 191 120 60 25 42 74 148)(22 190 109 59 26 41 75 147)(23 189 110 58 27 40 76 146)(24 188 111 57 28 39 77 145)
(1 111 68 77)(2 112 69 78)(3 113 70 79)(4 114 71 80)(5 115 72 81)(6 116 61 82)(7 117 62 83)(8 118 63 84)(9 119 64 73)(10 120 65 74)(11 109 66 75)(12 110 67 76)(13 137 29 131)(14 138 30 132)(15 139 31 121)(16 140 32 122)(17 141 33 123)(18 142 34 124)(19 143 35 125)(20 144 36 126)(21 133 25 127)(22 134 26 128)(23 135 27 129)(24 136 28 130)(37 91 186 162)(38 92 187 163)(39 93 188 164)(40 94 189 165)(41 95 190 166)(42 96 191 167)(43 85 192 168)(44 86 181 157)(45 87 182 158)(46 88 183 159)(47 89 184 160)(48 90 185 161)(49 175 149 101)(50 176 150 102)(51 177 151 103)(52 178 152 104)(53 179 153 105)(54 180 154 106)(55 169 155 107)(56 170 156 108)(57 171 145 97)(58 172 146 98)(59 173 147 99)(60 174 148 100)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E···4I4J4K6A6B6C8A8B8C8D12A12B12C12D12E···12P
order1222344444···44466688881212121212···12
size1111222224···424242221212121222224···4

39 irreducible representations

dim11111122222222222444
type++++++++++++-+---
imageC1C2C2C2C2C2S3D4D4D6D6D6Q16C4○D4C3⋊D4D12C4○D12C8.C22C3⋊Q16Q8.14D6
kernelC127Q16C12⋊C8C6.SD16C122Q8C2×C3⋊Q16Q8×C12C4×Q8C2×C12C3×Q8C42C4⋊C4C2×Q8C12C12C2×C4Q8C4C6C4C2
# reps11212112211142444122

Matrix representation of C127Q16 in GL4(𝔽73) generated by

596600
76600
00720
00072
,
473900
652600
00041
001641
,
431300
603000
00116
00972
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,72,0,0,0,0,72],[47,65,0,0,39,26,0,0,0,0,0,16,0,0,41,41],[43,60,0,0,13,30,0,0,0,0,1,9,0,0,16,72] >;

C127Q16 in GAP, Magma, Sage, TeX

C_{12}\rtimes_7Q_{16}
% in TeX

G:=Group("C12:7Q16");
// GroupNames label

G:=SmallGroup(192,590);
// by ID

G=gap.SmallGroup(192,590);
# by ID

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

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

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