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G = C12⋊(C4⋊C4)  order 192 = 26·3

1st semidirect product of C12 and C4⋊C4 acting via C4⋊C4/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C121(C4⋊C4), C6.57(C4×D4), C6.19(C4×Q8), C4⋊Dic312C4, C2.3(C12⋊Q8), C6.19(C4⋊Q8), (C2×C12).17Q8, C41(Dic3⋊C4), (C2×C12).136D4, (C2×C4).28Dic6, C2.3(D63D4), C22.18(S3×Q8), C6.87(C4⋊D4), C2.1(D63Q8), (C2×Dic3).14Q8, (C22×C4).347D6, C22.104(S3×D4), C6.69(C22⋊Q8), (C2×Dic3).107D4, C2.3(C4.Dic6), C6.15(C42.C2), C2.17(Dic35D4), C22.26(C2×Dic6), C2.9(Dic6⋊C4), C6.C42.14C2, (C22×C6).336C23, C23.296(C22×S3), C22.51(D42S3), (C22×C12).140C22, C33(C23.65C23), C22.21(Q83S3), (C22×Dic3).48C22, (C6×C4⋊C4).9C2, C6.35(C2×C4⋊C4), (C2×C4⋊C4).10S3, (C2×C4).77(C4×S3), (C2×C6).72(C2×Q8), (C2×C4×Dic3).3C2, (C2×C12).80(C2×C4), (C2×C6).327(C2×D4), C22.130(S3×C2×C4), C2.10(C2×Dic3⋊C4), (C2×C4⋊Dic3).31C2, C22.60(C2×C3⋊D4), (C2×C6).186(C4○D4), (C2×C4).183(C3⋊D4), (C2×Dic3⋊C4).11C2, (C2×C6).112(C22×C4), (C2×Dic3).28(C2×C4), SmallGroup(192,531)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12⋊(C4⋊C4)
C1C3C6C2×C6C22×C6C22×Dic3C2×C4×Dic3 — C12⋊(C4⋊C4)
C3C2×C6 — C12⋊(C4⋊C4)
C1C23C2×C4⋊C4

Generators and relations for C12⋊(C4⋊C4)
 G = < a,b,c | a12=b4=c4=1, bab-1=a-1, cac-1=a7, cbc-1=b-1 >

Subgroups: 376 in 170 conjugacy classes, 83 normal (41 characteristic)
C1, C2 [×7], C3, C4 [×4], C4 [×10], C22 [×7], C6 [×7], C2×C4 [×6], C2×C4 [×22], C23, Dic3 [×8], C12 [×4], C12 [×2], C2×C6 [×7], C42 [×2], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×6], C2×C12 [×6], C22×C6, C2.C42 [×2], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×4], C3×C4⋊C4 [×2], C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C23.65C23, C6.C42 [×2], C2×C4×Dic3, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C6×C4⋊C4, C12⋊(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], Q8 [×4], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C23.65C23, Dic6⋊C4, C12⋊Q8, C4.Dic6, Dic35D4, C2×Dic3⋊C4, D63D4, D63Q8, C12⋊(C4⋊C4)

Smallest permutation representation of C12⋊(C4⋊C4)
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 28 75 139)(2 27 76 138)(3 26 77 137)(4 25 78 136)(5 36 79 135)(6 35 80 134)(7 34 81 133)(8 33 82 144)(9 32 83 143)(10 31 84 142)(11 30 73 141)(12 29 74 140)(13 161 177 130)(14 160 178 129)(15 159 179 128)(16 158 180 127)(17 157 169 126)(18 168 170 125)(19 167 171 124)(20 166 172 123)(21 165 173 122)(22 164 174 121)(23 163 175 132)(24 162 176 131)(37 117 152 92)(38 116 153 91)(39 115 154 90)(40 114 155 89)(41 113 156 88)(42 112 145 87)(43 111 146 86)(44 110 147 85)(45 109 148 96)(46 120 149 95)(47 119 150 94)(48 118 151 93)(49 71 103 188)(50 70 104 187)(51 69 105 186)(52 68 106 185)(53 67 107 184)(54 66 108 183)(55 65 97 182)(56 64 98 181)(57 63 99 192)(58 62 100 191)(59 61 101 190)(60 72 102 189)
(1 19 38 62)(2 14 39 69)(3 21 40 64)(4 16 41 71)(5 23 42 66)(6 18 43 61)(7 13 44 68)(8 20 45 63)(9 15 46 70)(10 22 47 65)(11 17 48 72)(12 24 37 67)(25 127 113 49)(26 122 114 56)(27 129 115 51)(28 124 116 58)(29 131 117 53)(30 126 118 60)(31 121 119 55)(32 128 120 50)(33 123 109 57)(34 130 110 52)(35 125 111 59)(36 132 112 54)(73 169 151 189)(74 176 152 184)(75 171 153 191)(76 178 154 186)(77 173 155 181)(78 180 156 188)(79 175 145 183)(80 170 146 190)(81 177 147 185)(82 172 148 192)(83 179 149 187)(84 174 150 182)(85 106 133 161)(86 101 134 168)(87 108 135 163)(88 103 136 158)(89 98 137 165)(90 105 138 160)(91 100 139 167)(92 107 140 162)(93 102 141 157)(94 97 142 164)(95 104 143 159)(96 99 144 166)

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

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

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

48 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T6A···6G12A···12L
order12···23444444444···444446···612···12
size11···12222244446···6121212122···24···4

48 irreducible representations

dim111111122222222224444
type++++++++-+-+-+--+
imageC1C2C2C2C2C2C4S3D4Q8D4Q8D6C4○D4Dic6C4×S3C3⋊D4S3×D4D42S3S3×Q8Q83S3
kernelC12⋊(C4⋊C4)C6.C42C2×C4×Dic3C2×Dic3⋊C4C2×C4⋊Dic3C6×C4⋊C4C4⋊Dic3C2×C4⋊C4C2×Dic3C2×Dic3C2×C12C2×C12C22×C4C2×C6C2×C4C2×C4C2×C4C22C22C22C22
# reps121211812222344441111

Matrix representation of C12⋊(C4⋊C4) in GL7(𝔽13)

12000000
012110000
0110000
0001000
0000100
00000112
0000010
,
1000000
01200000
0110000
00031200
000101000
00000012
00000120
,
5000000
01200000
0110000
000121100
0000100
0000010
0000001

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,11,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,10,0,0,0,0,0,12,10,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0],[5,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

C12⋊(C4⋊C4) in GAP, Magma, Sage, TeX

C_{12}\rtimes (C_4\rtimes C_4)
% in TeX

G:=Group("C12:(C4:C4)");
// GroupNames label

G:=SmallGroup(192,531);
// by ID

G=gap.SmallGroup(192,531);
# by ID

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

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

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