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G = C4.(D6⋊C4)  order 192 = 26·3

9th non-split extension by C4 of D6⋊C4 acting via D6⋊C4/C2×Dic3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.20(C4×Q8), C4.9(D6⋊C4), (C2×Dic3)⋊6Q8, C6.24(C4⋊Q8), (C2×Dic6)⋊10C4, (C2×C12).137D4, (C2×C4).141D12, C22.19(S3×Q8), C2.3(C4.D12), (C22×C4).348D6, C22.45(C2×D12), C6.43(C22⋊Q8), C12.22(C22⋊C4), C6.39(C4.4D4), C2.1(Dic3⋊Q8), C2.2(C23.12D6), C6.C42.15C2, (C22×C6).337C23, C23.297(C22×S3), (C22×Dic6).11C2, C2.10(Dic6⋊C4), C22.52(D42S3), (C22×C12).141C22, C33(C23.67C23), (C22×Dic3).49C22, (C6×C4⋊C4).10C2, (C2×C4⋊C4).11S3, (C2×C4).78(C4×S3), C2.14(C2×D6⋊C4), (C2×C6).73(C2×Q8), (C2×C4×Dic3).4C2, (C2×C12).81(C2×C4), (C2×C6).150(C2×D4), C6.41(C2×C22⋊C4), C22.131(S3×C2×C4), C22.61(C2×C3⋊D4), (C2×C6).149(C4○D4), (C2×C4).126(C3⋊D4), (C2×C6).113(C22×C4), (C2×Dic3).29(C2×C4), SmallGroup(192,532)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4.(D6⋊C4)
C1C3C6C2×C6C22×C6C22×Dic3C2×C4×Dic3 — C4.(D6⋊C4)
C3C2×C6 — C4.(D6⋊C4)
C1C23C2×C4⋊C4

Generators and relations for C4.(D6⋊C4)
 G = < a,b,c,d | a4=b6=d4=1, c2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 440 in 186 conjugacy classes, 83 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×10], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×22], Q8 [×8], C23, Dic3 [×8], C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C42 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×8], Dic6 [×8], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×6], C2×C12 [×6], C22×C6, C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, C4×Dic3 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12, C22×C12 [×2], C23.67C23, C6.C42 [×4], C2×C4×Dic3, C6×C4⋊C4, C22×Dic6, C4.(D6⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], Q8 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, D6⋊C4 [×4], S3×C2×C4, C2×D12, D42S3 [×2], S3×Q8 [×2], C2×C3⋊D4, C23.67C23, Dic6⋊C4 [×2], C4.D12 [×2], C2×D6⋊C4, C23.12D6, Dic3⋊Q8, C4.(D6⋊C4)

Smallest permutation representation of C4.(D6⋊C4)
Regular action on 192 points
Generators in S192
(1 66 8 56)(2 61 9 57)(3 62 10 58)(4 63 11 59)(5 64 12 60)(6 65 7 55)(13 146 190 142)(14 147 191 143)(15 148 192 144)(16 149 187 139)(17 150 188 140)(18 145 189 141)(19 78 30 67)(20 73 25 68)(21 74 26 69)(22 75 27 70)(23 76 28 71)(24 77 29 72)(31 89 41 79)(32 90 42 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)(43 101 54 91)(44 102 49 92)(45 97 50 93)(46 98 51 94)(47 99 52 95)(48 100 53 96)(103 162 114 151)(104 157 109 152)(105 158 110 153)(106 159 111 154)(107 160 112 155)(108 161 113 156)(115 173 125 163)(116 174 126 164)(117 169 121 165)(118 170 122 166)(119 171 123 167)(120 172 124 168)(127 186 138 175)(128 181 133 176)(129 182 134 177)(130 183 135 178)(131 184 136 179)(132 185 137 180)
(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)
(1 145 8 141)(2 150 9 140)(3 149 10 139)(4 148 11 144)(5 147 12 143)(6 146 7 142)(13 55 190 65)(14 60 191 64)(15 59 192 63)(16 58 187 62)(17 57 188 61)(18 56 189 66)(19 138 30 127)(20 137 25 132)(21 136 26 131)(22 135 27 130)(23 134 28 129)(24 133 29 128)(31 103 41 114)(32 108 42 113)(33 107 37 112)(34 106 38 111)(35 105 39 110)(36 104 40 109)(43 118 54 122)(44 117 49 121)(45 116 50 126)(46 115 51 125)(47 120 52 124)(48 119 53 123)(67 175 78 186)(68 180 73 185)(69 179 74 184)(70 178 75 183)(71 177 76 182)(72 176 77 181)(79 162 89 151)(80 161 90 156)(81 160 85 155)(82 159 86 154)(83 158 87 153)(84 157 88 152)(91 170 101 166)(92 169 102 165)(93 174 97 164)(94 173 98 163)(95 172 99 168)(96 171 100 167)
(1 49 25 32)(2 50 26 33)(3 51 27 34)(4 52 28 35)(5 53 29 36)(6 54 30 31)(7 43 19 41)(8 44 20 42)(9 45 21 37)(10 46 22 38)(11 47 23 39)(12 48 24 40)(13 163 175 159)(14 164 176 160)(15 165 177 161)(16 166 178 162)(17 167 179 157)(18 168 180 158)(55 91 78 89)(56 92 73 90)(57 93 74 85)(58 94 75 86)(59 95 76 87)(60 96 77 88)(61 97 69 81)(62 98 70 82)(63 99 71 83)(64 100 72 84)(65 101 67 79)(66 102 68 80)(103 149 122 130)(104 150 123 131)(105 145 124 132)(106 146 125 127)(107 147 126 128)(108 148 121 129)(109 140 119 136)(110 141 120 137)(111 142 115 138)(112 143 116 133)(113 144 117 134)(114 139 118 135)(151 187 170 183)(152 188 171 184)(153 189 172 185)(154 190 173 186)(155 191 174 181)(156 192 169 182)

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

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

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

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

dim1111112222222244
type++++++-+++--
imageC1C2C2C2C2C4S3Q8D4D6C4○D4C4×S3D12C3⋊D4D42S3S3×Q8
kernelC4.(D6⋊C4)C6.C42C2×C4×Dic3C6×C4⋊C4C22×Dic6C2×Dic6C2×C4⋊C4C2×Dic3C2×C12C22×C4C2×C6C2×C4C2×C4C2×C4C22C22
# reps1411181443444422

Matrix representation of C4.(D6⋊C4) in GL6(𝔽13)

390000
9100000
001000
000100
000010
000001
,
100000
010000
0012000
0001200
0000121
0000120
,
0120000
100000
0012000
0010100
000001
000010
,
010000
1200000
005100
002800
000080
000008

G:=sub<GL(6,GF(13))| [3,9,0,0,0,0,9,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,10,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,2,0,0,0,0,1,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

C4.(D6⋊C4) in GAP, Magma, Sage, TeX

C_4.(D_6\rtimes C_4)
% in TeX

G:=Group("C4.(D6:C4)");
// GroupNames label

G:=SmallGroup(192,532);
// by ID

G=gap.SmallGroup(192,532);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,310,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^6=d^4=1,c^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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