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G = C4⋊C4⋊6Dic3  order 192 = 26·3

4th semidirect product of C4⋊C4 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12⋊3(C4⋊C4), C4⋊C4⋊6Dic3, C6.89(C4×D4), C6.25(C4×Q8), C4⋊1(C4⋊Dic3), C2.5(C12⋊Q8), C6.23(C4⋊Q8), (C2×C12).20Q8, C2.4(Q8×Dic3), C2.7(D4×Dic3), (C2×C4).142D12, (C2×C12).140D4, (C2×C4).31Dic6, C2.3(C12⋊D4), C22.25(S3×Q8), C6.52(C4⋊D4), C2.4(C4.D12), (C2×Dic3).19Q8, C22.46(C2×D12), (C22×C4).117D6, C22.109(S3×D4), C6.48(C22⋊Q8), (C2×Dic3).109D4, C2.5(C4.Dic6), C6.23(C42.C2), C22.29(C2×Dic6), C6.C42.29C2, C23.304(C22×S3), (C22×C12).66C22, (C22×C6).348C23, C22.58(D4⋊2S3), C3⋊5(C23.65C23), C22.26(Q8⋊3S3), C22.42(C22×Dic3), (C22×Dic3).191C22, (C3×C4⋊C4)⋊9C4, C6.38(C2×C4⋊C4), (C6×C4⋊C4).15C2, (C2×C4⋊C4).21S3, C2.8(C2×C4⋊Dic3), (C2×C6).38(C2×Q8), (C2×C4×Dic3).8C2, (C2×C12).45(C2×C4), (C2×C6).333(C2×D4), (C2×C4⋊Dic3).35C2, (C2×C4).18(C2×Dic3), (C2×C6).187(C4○D4), (C2×C6).181(C22×C4), SmallGroup(192,543)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C4⋊C4⋊6Dic3
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C4⋊C4⋊6Dic3
Lower central
C3 — C2×C6 — C4⋊C4⋊6Dic3
Upper central
C1 — C23 — C2×C4⋊C4

Generators and relations for C4⋊C4⋊6Dic3
 G = < a,b,c,d | a4=b4=c6=1, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 376 in 170 conjugacy classes, 91 normal (41 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Dic3, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23.65C23, C6.C42, C2×C4×Dic3, C2×C4⋊Dic3, C2×C4⋊Dic3, C6×C4⋊C4, C4⋊C4⋊6Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Dic3, C2×Dic6, C2×D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×Dic3, C23.65C23, C12⋊Q8, C4.Dic6, C12⋊D4, C4.D12, C2×C4⋊Dic3, D4×Dic3, Q8×Dic3, C4⋊C4⋊6Dic3

Smallest permutation representation of C4⋊C4⋊6Dic3
►Regular action on 192 points
Generators in S192
(1 76 23 62)(2 77 24 63)(3 78 19 64)(4 73 20 65)(5 74 21 66)(6 75 22 61)(7 72 29 59)(8 67 30 60)(9 68 25 55)(10 69 26 56)(11 70 27 57)(12 71 28 58)(13 142 181 128)(14 143 182 129)(15 144 183 130)(16 139 184 131)(17 140 185 132)(18 141 186 127)(31 102 44 89)(32 97 45 90)(33 98 46 85)(34 99 47 86)(35 100 48 87)(36 101 43 88)(37 94 51 81)(38 95 52 82)(39 96 53 83)(40 91 54 84)(41 92 49 79)(42 93 50 80)(103 169 117 162)(104 170 118 157)(105 171 119 158)(106 172 120 159)(107 173 115 160)(108 174 116 161)(109 166 122 152)(110 167 123 153)(111 168 124 154)(112 163 125 155)(113 164 126 156)(114 165 121 151)(133 190 146 176)(134 191 147 177)(135 192 148 178)(136 187 149 179)(137 188 150 180)(138 189 145 175)

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

(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)

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

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

dim11111122222222224444
type+++++++-+--+-++--+
imageC1C2C2C2C2C4S3D4Q8D4Q8Dic3D6C4○D4Dic6D12S3×D4D4⋊2S3S3×Q8Q8⋊3S3
kernelC4⋊C4⋊6Dic3C6.C42C2×C4×Dic3C2×C4⋊Dic3C6×C4⋊C4C3×C4⋊C4C2×C4⋊C4C2×Dic3C2×Dic3C2×C12C2×C12C4⋊C4C22×C4C2×C6C2×C4C2×C4C22C22C22C22
# reps12131812222434441111

Matrix representation of C4⋊C4⋊6Dic3 ►in GL8(𝔽13)

68000000
107000000
001200000
000120000
00005000
00000800
00000010
00000001
,
111000000
112000000
0012110000
00110000
00000100
000012000
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
00000030
00000029
,
68000000
107000000
00590000
00680000
000012000
00000100
00000052
00000018

G:=sub<GL(8,GF(13))| [6,10,0,0,0,0,0,0,8,7,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,9],[6,10,0,0,0,0,0,0,8,7,0,0,0,0,0,0,0,0,5,6,0,0,0,0,0,0,9,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,2,8] >;
 

C4⋊C4⋊6Dic3 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4\rtimes_6{\rm Dic}_3
 
% in TeX
 
G:=Group("C4:C4:6Dic3");
 
// GroupNames label
 
G:=SmallGroup(192,543);
 
// by ID
 
G=gap.SmallGroup(192,543);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,100,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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