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## G = C42⋊10Dic3order 192 = 26·3

### 5th semidirect product of C42 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊10Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — C42⋊10Dic3
 Lower central C3 — C2×C6 — C42⋊10Dic3
 Upper central C1 — C23 — C2×C42

Generators and relations for C4210Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 408 in 186 conjugacy classes, 119 normal (11 characteristic)
C1, C2, C2 [×6], C3, C4 [×12], C4 [×4], C22, C22 [×6], C6, C6 [×6], C2×C4 [×18], C2×C4 [×12], C23, Dic3 [×4], C12 [×12], C2×C6, C2×C6 [×6], C42 [×4], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×12], C2×C12 [×18], C22×C6, C2×C42, C2×C4⋊C4 [×6], C4⋊Dic3 [×12], C4×C12 [×4], C22×Dic3 [×4], C22×C12 [×3], C429C4, C2×C4⋊Dic3 [×6], C2×C4×C12, C4210Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×6], Q8 [×6], C23, Dic3 [×4], D6 [×3], C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], Dic6 [×6], D12 [×6], C2×Dic3 [×6], C22×S3, C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C4⋊Dic3 [×12], C2×Dic6 [×3], C2×D12 [×3], C22×Dic3, C429C4, C122Q8 [×3], C4⋊D12, C2×C4⋊Dic3 [×3], C4210Dic3

Smallest permutation representation of C4210Dic3
Regular action on 192 points
Generators in S192
(1 71 23 56)(2 72 24 57)(3 67 19 58)(4 68 20 59)(5 69 21 60)(6 70 22 55)(7 75 27 65)(8 76 28 66)(9 77 29 61)(10 78 30 62)(11 73 25 63)(12 74 26 64)(13 135 183 146)(14 136 184 147)(15 137 185 148)(16 138 186 149)(17 133 181 150)(18 134 182 145)(31 94 46 79)(32 95 47 80)(33 96 48 81)(34 91 43 82)(35 92 44 83)(36 93 45 84)(37 100 53 85)(38 101 54 86)(39 102 49 87)(40 97 50 88)(41 98 51 89)(42 99 52 90)(103 151 115 163)(104 152 116 164)(105 153 117 165)(106 154 118 166)(107 155 119 167)(108 156 120 168)(109 157 126 174)(110 158 121 169)(111 159 122 170)(112 160 123 171)(113 161 124 172)(114 162 125 173)(127 175 139 187)(128 176 140 188)(129 177 141 189)(130 178 142 190)(131 179 143 191)(132 180 144 192)
(1 39 8 35)(2 40 9 36)(3 41 10 31)(4 42 11 32)(5 37 12 33)(6 38 7 34)(13 163 190 173)(14 164 191 174)(15 165 192 169)(16 166 187 170)(17 167 188 171)(18 168 189 172)(19 51 30 46)(20 52 25 47)(21 53 26 48)(22 54 27 43)(23 49 28 44)(24 50 29 45)(55 86 65 82)(56 87 66 83)(57 88 61 84)(58 89 62 79)(59 90 63 80)(60 85 64 81)(67 98 78 94)(68 99 73 95)(69 100 74 96)(70 101 75 91)(71 102 76 92)(72 97 77 93)(103 130 114 135)(104 131 109 136)(105 132 110 137)(106 127 111 138)(107 128 112 133)(108 129 113 134)(115 142 125 146)(116 143 126 147)(117 144 121 148)(118 139 122 149)(119 140 123 150)(120 141 124 145)(151 178 162 183)(152 179 157 184)(153 180 158 185)(154 175 159 186)(155 176 160 181)(156 177 161 182)
(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)
(1 106 4 103)(2 105 5 108)(3 104 6 107)(7 112 10 109)(8 111 11 114)(9 110 12 113)(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)| (1,71,23,56)(2,72,24,57)(3,67,19,58)(4,68,20,59)(5,69,21,60)(6,70,22,55)(7,75,27,65)(8,76,28,66)(9,77,29,61)(10,78,30,62)(11,73,25,63)(12,74,26,64)(13,135,183,146)(14,136,184,147)(15,137,185,148)(16,138,186,149)(17,133,181,150)(18,134,182,145)(31,94,46,79)(32,95,47,80)(33,96,48,81)(34,91,43,82)(35,92,44,83)(36,93,45,84)(37,100,53,85)(38,101,54,86)(39,102,49,87)(40,97,50,88)(41,98,51,89)(42,99,52,90)(103,151,115,163)(104,152,116,164)(105,153,117,165)(106,154,118,166)(107,155,119,167)(108,156,120,168)(109,157,126,174)(110,158,121,169)(111,159,122,170)(112,160,123,171)(113,161,124,172)(114,162,125,173)(127,175,139,187)(128,176,140,188)(129,177,141,189)(130,178,142,190)(131,179,143,191)(132,180,144,192), (1,39,8,35)(2,40,9,36)(3,41,10,31)(4,42,11,32)(5,37,12,33)(6,38,7,34)(13,163,190,173)(14,164,191,174)(15,165,192,169)(16,166,187,170)(17,167,188,171)(18,168,189,172)(19,51,30,46)(20,52,25,47)(21,53,26,48)(22,54,27,43)(23,49,28,44)(24,50,29,45)(55,86,65,82)(56,87,66,83)(57,88,61,84)(58,89,62,79)(59,90,63,80)(60,85,64,81)(67,98,78,94)(68,99,73,95)(69,100,74,96)(70,101,75,91)(71,102,76,92)(72,97,77,93)(103,130,114,135)(104,131,109,136)(105,132,110,137)(106,127,111,138)(107,128,112,133)(108,129,113,134)(115,142,125,146)(116,143,126,147)(117,144,121,148)(118,139,122,149)(119,140,123,150)(120,141,124,145)(151,178,162,183)(152,179,157,184)(153,180,158,185)(154,175,159,186)(155,176,160,181)(156,177,161,182), (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), (1,106,4,103)(2,105,5,108)(3,104,6,107)(7,112,10,109)(8,111,11,114)(9,110,12,113)(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( (1,71,23,56)(2,72,24,57)(3,67,19,58)(4,68,20,59)(5,69,21,60)(6,70,22,55)(7,75,27,65)(8,76,28,66)(9,77,29,61)(10,78,30,62)(11,73,25,63)(12,74,26,64)(13,135,183,146)(14,136,184,147)(15,137,185,148)(16,138,186,149)(17,133,181,150)(18,134,182,145)(31,94,46,79)(32,95,47,80)(33,96,48,81)(34,91,43,82)(35,92,44,83)(36,93,45,84)(37,100,53,85)(38,101,54,86)(39,102,49,87)(40,97,50,88)(41,98,51,89)(42,99,52,90)(103,151,115,163)(104,152,116,164)(105,153,117,165)(106,154,118,166)(107,155,119,167)(108,156,120,168)(109,157,126,174)(110,158,121,169)(111,159,122,170)(112,160,123,171)(113,161,124,172)(114,162,125,173)(127,175,139,187)(128,176,140,188)(129,177,141,189)(130,178,142,190)(131,179,143,191)(132,180,144,192), (1,39,8,35)(2,40,9,36)(3,41,10,31)(4,42,11,32)(5,37,12,33)(6,38,7,34)(13,163,190,173)(14,164,191,174)(15,165,192,169)(16,166,187,170)(17,167,188,171)(18,168,189,172)(19,51,30,46)(20,52,25,47)(21,53,26,48)(22,54,27,43)(23,49,28,44)(24,50,29,45)(55,86,65,82)(56,87,66,83)(57,88,61,84)(58,89,62,79)(59,90,63,80)(60,85,64,81)(67,98,78,94)(68,99,73,95)(69,100,74,96)(70,101,75,91)(71,102,76,92)(72,97,77,93)(103,130,114,135)(104,131,109,136)(105,132,110,137)(106,127,111,138)(107,128,112,133)(108,129,113,134)(115,142,125,146)(116,143,126,147)(117,144,121,148)(118,139,122,149)(119,140,123,150)(120,141,124,145)(151,178,162,183)(152,179,157,184)(153,180,158,185)(154,175,159,186)(155,176,160,181)(156,177,161,182), (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), (1,106,4,103)(2,105,5,108)(3,104,6,107)(7,112,10,109)(8,111,11,114)(9,110,12,113)(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([(1,71,23,56),(2,72,24,57),(3,67,19,58),(4,68,20,59),(5,69,21,60),(6,70,22,55),(7,75,27,65),(8,76,28,66),(9,77,29,61),(10,78,30,62),(11,73,25,63),(12,74,26,64),(13,135,183,146),(14,136,184,147),(15,137,185,148),(16,138,186,149),(17,133,181,150),(18,134,182,145),(31,94,46,79),(32,95,47,80),(33,96,48,81),(34,91,43,82),(35,92,44,83),(36,93,45,84),(37,100,53,85),(38,101,54,86),(39,102,49,87),(40,97,50,88),(41,98,51,89),(42,99,52,90),(103,151,115,163),(104,152,116,164),(105,153,117,165),(106,154,118,166),(107,155,119,167),(108,156,120,168),(109,157,126,174),(110,158,121,169),(111,159,122,170),(112,160,123,171),(113,161,124,172),(114,162,125,173),(127,175,139,187),(128,176,140,188),(129,177,141,189),(130,178,142,190),(131,179,143,191),(132,180,144,192)], [(1,39,8,35),(2,40,9,36),(3,41,10,31),(4,42,11,32),(5,37,12,33),(6,38,7,34),(13,163,190,173),(14,164,191,174),(15,165,192,169),(16,166,187,170),(17,167,188,171),(18,168,189,172),(19,51,30,46),(20,52,25,47),(21,53,26,48),(22,54,27,43),(23,49,28,44),(24,50,29,45),(55,86,65,82),(56,87,66,83),(57,88,61,84),(58,89,62,79),(59,90,63,80),(60,85,64,81),(67,98,78,94),(68,99,73,95),(69,100,74,96),(70,101,75,91),(71,102,76,92),(72,97,77,93),(103,130,114,135),(104,131,109,136),(105,132,110,137),(106,127,111,138),(107,128,112,133),(108,129,113,134),(115,142,125,146),(116,143,126,147),(117,144,121,148),(118,139,122,149),(119,140,123,150),(120,141,124,145),(151,178,162,183),(152,179,157,184),(153,180,158,185),(154,175,159,186),(155,176,160,181),(156,177,161,182)], [(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)], [(1,106,4,103),(2,105,5,108),(3,104,6,107),(7,112,10,109),(8,111,11,114),(9,110,12,113),(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)])

60 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4L 4M ··· 4T 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C4 S3 D4 Q8 Dic3 D6 Dic6 D12 kernel C42⋊10Dic3 C2×C4⋊Dic3 C2×C4×C12 C4×C12 C2×C42 C2×C12 C2×C12 C42 C22×C4 C2×C4 C2×C4 # reps 1 6 1 8 1 6 6 4 3 12 12

Matrix representation of C4210Dic3 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 10 7 0 0 0 6 3
,
 1 0 0 0 0 0 10 6 0 0 0 7 3 0 0 0 0 0 3 6 0 0 0 7 10
,
 12 0 0 0 0 0 12 1 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 1 1
,
 5 0 0 0 0 0 7 3 0 0 0 10 6 0 0 0 0 0 9 11 0 0 0 2 4

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,6,0,0,0,7,3],[1,0,0,0,0,0,10,7,0,0,0,6,3,0,0,0,0,0,3,7,0,0,0,6,10],[12,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,12,1],[5,0,0,0,0,0,7,10,0,0,0,3,6,0,0,0,0,0,9,2,0,0,0,11,4] >;

C4210Dic3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{10}{\rm Dic}_3
% in TeX

G:=Group("C4^2:10Dic3");
// GroupNames label

G:=SmallGroup(192,494);
// by ID

G=gap.SmallGroup(192,494);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,232,422,100,6278]);
// Polycyclic

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

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