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G = C124(C4⋊C4)  order 192 = 26·3

1st semidirect product of C12 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C124(C4⋊C4), C6.8(C4×Q8), C4⋊Dic39C4, C6.15(C4×D4), C6.6(C4⋊Q8), C2.18(C4×D12), (C2×C4).65D12, (C2×C12).52Q8, C43(Dic3⋊C4), (C2×C12).469D4, (C2×C42).14S3, C2.10(C4×Dic6), (C2×C4).40Dic6, C6.55(C4⋊D4), C2.1(C127D4), C2.1(C122Q8), (C22×C4).410D6, C22.33(C2×D12), C6.52(C22⋊Q8), C6.2(C42.C2), C6.C42.9C2, C2.1(C12.6Q8), C22.16(C2×Dic6), C2.1(C12.48D4), C22.39(C4○D12), C23.271(C22×S3), (C22×C6).303C23, (C22×C12).469C22, C32(C23.65C23), (C22×Dic3).26C22, (C2×C4×C12).9C2, C6.24(C2×C4⋊C4), (C2×C6).23(C2×Q8), (C2×C4).108(C4×S3), (C2×C6).423(C2×D4), C2.4(C2×Dic3⋊C4), C22.116(S3×C2×C4), (C2×C12).223(C2×C4), (C2×C6).64(C4○D4), (C2×Dic3⋊C4).9C2, (C2×C4⋊Dic3).13C2, (C2×C6).93(C22×C4), C22.38(C2×C3⋊D4), (C2×C4).237(C3⋊D4), (C2×Dic3).24(C2×C4), SmallGroup(192,487)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C124(C4⋊C4)
C1C3C6C2×C6C22×C6C22×Dic3C2×C4⋊Dic3 — C124(C4⋊C4)
C3C2×C6 — C124(C4⋊C4)
C1C23C2×C42

Generators and relations for C124(C4⋊C4)
 G = < a,b,c | a12=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 376 in 170 conjugacy classes, 87 normal (43 characteristic)
C1, C2 [×7], C3, C4 [×4], C4 [×10], C22 [×7], C6 [×7], C2×C4 [×10], C2×C4 [×18], C23, Dic3 [×6], C12 [×4], C12 [×4], C2×C6 [×7], C42 [×2], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×10], C2×C12 [×4], C22×C6, C2.C42 [×2], C2×C42, C2×C4⋊C4 [×4], Dic3⋊C4 [×4], C4⋊Dic3 [×4], C4⋊Dic3 [×2], C4×C12 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.65C23, C6.C42 [×2], C2×Dic3⋊C4 [×2], C2×C4⋊Dic3 [×2], C2×C4×C12, C124(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 [×4], C4×S3 [×2], D12 [×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 [×2], S3×C2×C4, C2×D12, C4○D12 [×2], C2×C3⋊D4, C23.65C23, C4×Dic6, C122Q8, C12.6Q8, C4×D12, C2×Dic3⋊C4, C12.48D4, C127D4, C124(C4⋊C4)

Smallest permutation representation of C124(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 94 136 98)(2 93 137 97)(3 92 138 108)(4 91 139 107)(5 90 140 106)(6 89 141 105)(7 88 142 104)(8 87 143 103)(9 86 144 102)(10 85 133 101)(11 96 134 100)(12 95 135 99)(13 181 82 32)(14 192 83 31)(15 191 84 30)(16 190 73 29)(17 189 74 28)(18 188 75 27)(19 187 76 26)(20 186 77 25)(21 185 78 36)(22 184 79 35)(23 183 80 34)(24 182 81 33)(37 125 120 175)(38 124 109 174)(39 123 110 173)(40 122 111 172)(41 121 112 171)(42 132 113 170)(43 131 114 169)(44 130 115 180)(45 129 116 179)(46 128 117 178)(47 127 118 177)(48 126 119 176)(49 72 167 155)(50 71 168 154)(51 70 157 153)(52 69 158 152)(53 68 159 151)(54 67 160 150)(55 66 161 149)(56 65 162 148)(57 64 163 147)(58 63 164 146)(59 62 165 145)(60 61 166 156)
(1 114 15 161)(2 115 16 162)(3 116 17 163)(4 117 18 164)(5 118 19 165)(6 119 20 166)(7 120 21 167)(8 109 22 168)(9 110 23 157)(10 111 24 158)(11 112 13 159)(12 113 14 160)(25 156 105 176)(26 145 106 177)(27 146 107 178)(28 147 108 179)(29 148 97 180)(30 149 98 169)(31 150 99 170)(32 151 100 171)(33 152 101 172)(34 153 102 173)(35 154 103 174)(36 155 104 175)(37 78 49 142)(38 79 50 143)(39 80 51 144)(40 81 52 133)(41 82 53 134)(42 83 54 135)(43 84 55 136)(44 73 56 137)(45 74 57 138)(46 75 58 139)(47 76 59 140)(48 77 60 141)(61 89 126 186)(62 90 127 187)(63 91 128 188)(64 92 129 189)(65 93 130 190)(66 94 131 191)(67 95 132 192)(68 96 121 181)(69 85 122 182)(70 86 123 183)(71 87 124 184)(72 88 125 185)

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

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

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

60 conjugacy classes

class 1 2A···2G 3 4A···4L4M···4T6A···6G12A···12X
order12···234···44···46···612···12
size11···122···212···122···22···2

60 irreducible representations

dim1111112222222222
type+++++++-+-+
imageC1C2C2C2C2C4S3D4Q8D6C4○D4Dic6C4×S3D12C3⋊D4C4○D12
kernelC124(C4⋊C4)C6.C42C2×Dic3⋊C4C2×C4⋊Dic3C2×C4×C12C4⋊Dic3C2×C42C2×C12C2×C12C22×C4C2×C6C2×C4C2×C4C2×C4C2×C4C22
# reps1222181443484448

Matrix representation of C124(C4⋊C4) in GL5(𝔽13)

10000
01000
00100
00033
000106
,
10000
051000
00800
00037
0001010
,
50000
0121100
01100
00080
00008

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,10,0,0,0,3,6],[1,0,0,0,0,0,5,0,0,0,0,10,8,0,0,0,0,0,3,10,0,0,0,7,10],[5,0,0,0,0,0,12,1,0,0,0,11,1,0,0,0,0,0,8,0,0,0,0,0,8] >;

C124(C4⋊C4) in GAP, Magma, Sage, TeX

C_{12}\rtimes_4(C_4\rtimes C_4)
% in TeX

G:=Group("C12:4(C4:C4)");
// GroupNames label

G:=SmallGroup(192,487);
// by ID

G=gap.SmallGroup(192,487);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,758,58,6278]);
// Polycyclic

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

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