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G = C12.17D8order 192 = 26·3

17th non-split extension by C12 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.17D8, C12.8Q16, C42.220D6, C3⋊C87Q8, C4⋊Q8.4S3, C4⋊C4.79D6, C33(C82Q8), C6.59(C2×D8), C4.34(S3×Q8), C4.6(D4⋊S3), C6.30(C4⋊Q8), C6.40(C2×Q16), C12.35(C2×Q8), (C2×C12).151D4, C4.3(C3⋊Q16), C122Q8.18C2, C6.Q16.15C2, (C2×C12).396C23, (C4×C12).125C22, C4⋊Dic3.156C22, C2.10(Dic3⋊Q8), (C4×C3⋊C8).11C2, (C3×C4⋊Q8).4C2, C2.14(C2×D4⋊S3), (C2×C6).527(C2×D4), C2.11(C2×C3⋊Q16), (C2×C3⋊C8).259C22, (C2×C4).133(C3⋊D4), (C3×C4⋊C4).126C22, (C2×C4).493(C22×S3), C22.199(C2×C3⋊D4), SmallGroup(192,637)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C12.17D8
C1C3C6C2×C6C2×C12C2×C3⋊C8C4×C3⋊C8 — C12.17D8
C3C6C2×C12 — C12.17D8
C1C22C42C4⋊Q8

Generators and relations for C12.17D8
 G = < a,b,c | a12=b8=1, c2=a6, bab-1=a5, cac-1=a-1, cbc-1=b-1 >

Subgroups: 240 in 98 conjugacy classes, 51 normal (23 characteristic)
C1, C2 [×3], C3, C4 [×6], C4 [×4], C22, C6 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×4], Dic3 [×2], C12 [×6], C12 [×2], C2×C6, C42, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×Q8 [×2], C3⋊C8 [×4], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C4×C8, C2.D8 [×4], C4⋊Q8, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C6×Q8, C82Q8, C4×C3⋊C8, C6.Q16 [×4], C122Q8, C3×C4⋊Q8, C12.17D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], D8 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], C3⋊D4 [×2], C22×S3, C4⋊Q8, C2×D8, C2×Q16, D4⋊S3 [×2], C3⋊Q16 [×2], S3×Q8 [×2], C2×C3⋊D4, C82Q8, C2×D4⋊S3, C2×C3⋊Q16, Dic3⋊Q8, C12.17D8

Smallest permutation representation of C12.17D8
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 53 184 34 97 61 44 170)(2 58 185 27 98 66 45 175)(3 51 186 32 99 71 46 180)(4 56 187 25 100 64 47 173)(5 49 188 30 101 69 48 178)(6 54 189 35 102 62 37 171)(7 59 190 28 103 67 38 176)(8 52 191 33 104 72 39 169)(9 57 192 26 105 65 40 174)(10 50 181 31 106 70 41 179)(11 55 182 36 107 63 42 172)(12 60 183 29 108 68 43 177)(13 153 87 131 78 143 111 166)(14 146 88 124 79 136 112 159)(15 151 89 129 80 141 113 164)(16 156 90 122 81 134 114 157)(17 149 91 127 82 139 115 162)(18 154 92 132 83 144 116 167)(19 147 93 125 84 137 117 160)(20 152 94 130 73 142 118 165)(21 145 95 123 74 135 119 158)(22 150 96 128 75 140 120 163)(23 155 85 121 76 133 109 168)(24 148 86 126 77 138 110 161)
(1 83 7 77)(2 82 8 76)(3 81 9 75)(4 80 10 74)(5 79 11 73)(6 78 12 84)(13 108 19 102)(14 107 20 101)(15 106 21 100)(16 105 22 99)(17 104 23 98)(18 103 24 97)(25 151 31 145)(26 150 32 156)(27 149 33 155)(28 148 34 154)(29 147 35 153)(30 146 36 152)(37 111 43 117)(38 110 44 116)(39 109 45 115)(40 120 46 114)(41 119 47 113)(42 118 48 112)(49 124 55 130)(50 123 56 129)(51 122 57 128)(52 121 58 127)(53 132 59 126)(54 131 60 125)(61 167 67 161)(62 166 68 160)(63 165 69 159)(64 164 70 158)(65 163 71 157)(66 162 72 168)(85 185 91 191)(86 184 92 190)(87 183 93 189)(88 182 94 188)(89 181 95 187)(90 192 96 186)(133 175 139 169)(134 174 140 180)(135 173 141 179)(136 172 142 178)(137 171 143 177)(138 170 144 176)

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

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

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

36 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J6A6B6C8A···8H12A···12F12G12H12I12J
order122234···444446668···812···1212121212
size111122···28824242226···64···48888

36 irreducible representations

dim1111122222222444
type++++++-++++-+--
imageC1C2C2C2C2S3Q8D4D6D6D8Q16C3⋊D4D4⋊S3C3⋊Q16S3×Q8
kernelC12.17D8C4×C3⋊C8C6.Q16C122Q8C3×C4⋊Q8C4⋊Q8C3⋊C8C2×C12C42C4⋊C4C12C12C2×C4C4C4C4
# reps1141114212444222

Matrix representation of C12.17D8 in GL6(𝔽73)

0720000
100000
0007200
001100
000010
000001
,
16160000
57160000
00191400
00685400
00001616
00005716
,
19520000
52540000
00511000
00322200
00004314
00001430

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,57,0,0,0,0,16,16,0,0,0,0,0,0,19,68,0,0,0,0,14,54,0,0,0,0,0,0,16,57,0,0,0,0,16,16],[19,52,0,0,0,0,52,54,0,0,0,0,0,0,51,32,0,0,0,0,10,22,0,0,0,0,0,0,43,14,0,0,0,0,14,30] >;

C12.17D8 in GAP, Magma, Sage, TeX

C_{12}._{17}D_8
% in TeX

G:=Group("C12.17D8");
// GroupNames label

G:=SmallGroup(192,637);
// by ID

G=gap.SmallGroup(192,637);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,422,135,58,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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