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## G = C22×C4×C12order 192 = 26·3

### Abelian group of type [2,2,4,12]

Aliases: C22×C4×C12, SmallGroup(192,1400)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C4×C12
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C4×C12 — C2×C4×C12 — C22×C4×C12
 Lower central C1 — C22×C4×C12
 Upper central C1 — C22×C4×C12

Generators and relations for C22×C4×C12
G = < a,b,c,d | a2=b2=c4=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 498, all normal (8 characteristic)
C1, C2 [×15], C3, C4 [×24], C22, C22 [×34], C6 [×15], C2×C4 [×84], C23 [×15], C12 [×24], C2×C6, C2×C6 [×34], C42 [×16], C22×C4 [×42], C24, C2×C12 [×84], C22×C6 [×15], C2×C42 [×12], C23×C4 [×3], C4×C12 [×16], C22×C12 [×42], C23×C6, C22×C42, C2×C4×C12 [×12], C23×C12 [×3], C22×C4×C12
Quotients: C1, C2 [×15], C3, C4 [×24], C22 [×35], C6 [×15], C2×C4 [×84], C23 [×15], C12 [×24], C2×C6 [×35], C42 [×16], C22×C4 [×42], C24, C2×C12 [×84], C22×C6 [×15], C2×C42 [×12], C23×C4 [×3], C4×C12 [×16], C22×C12 [×42], C23×C6, C22×C42, C2×C4×C12 [×12], C23×C12 [×3], C22×C4×C12

Smallest permutation representation of C22×C4×C12
Regular action on 192 points
Generators in S192
(1 190)(2 191)(3 192)(4 181)(5 182)(6 183)(7 184)(8 185)(9 186)(10 187)(11 188)(12 189)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 168)(26 157)(27 158)(28 159)(29 160)(30 161)(31 162)(32 163)(33 164)(34 165)(35 166)(36 167)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 97)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 169)(61 129)(62 130)(63 131)(64 132)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 133)(117 134)(118 135)(119 136)(120 137)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(37 80)(38 81)(39 82)(40 83)(41 84)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(85 144)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(121 180)(122 169)(123 170)(124 171)(125 172)(126 173)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 181)(166 182)(167 183)(168 184)
(1 95 107 131)(2 96 108 132)(3 85 97 121)(4 86 98 122)(5 87 99 123)(6 88 100 124)(7 89 101 125)(8 90 102 126)(9 91 103 127)(10 92 104 128)(11 93 105 129)(12 94 106 130)(13 48 65 192)(14 37 66 181)(15 38 67 182)(16 39 68 183)(17 40 69 184)(18 41 70 185)(19 42 71 186)(20 43 72 187)(21 44 61 188)(22 45 62 189)(23 46 63 190)(24 47 64 191)(25 136 149 172)(26 137 150 173)(27 138 151 174)(28 139 152 175)(29 140 153 176)(30 141 154 177)(31 142 155 178)(32 143 156 179)(33 144 145 180)(34 133 146 169)(35 134 147 170)(36 135 148 171)(49 166 117 81)(50 167 118 82)(51 168 119 83)(52 157 120 84)(53 158 109 73)(54 159 110 74)(55 160 111 75)(56 161 112 76)(57 162 113 77)(58 163 114 78)(59 164 115 79)(60 165 116 80)
(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)

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

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

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

192 conjugacy classes

 class 1 2A ··· 2O 3A 3B 4A ··· 4AV 6A ··· 6AD 12A ··· 12CR order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

192 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C12 kernel C22×C4×C12 C2×C4×C12 C23×C12 C22×C42 C22×C12 C2×C42 C23×C4 C22×C4 # reps 1 12 3 2 48 24 6 96

Matrix representation of C22×C4×C12 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 8 0 0 0 0 1 0 0 0 0 12
,
 3 0 0 0 0 11 0 0 0 0 12 0 0 0 0 5
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,8,0,0,0,0,1,0,0,0,0,12],[3,0,0,0,0,11,0,0,0,0,12,0,0,0,0,5] >;

C22×C4×C12 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times C_{12}
% in TeX

G:=Group("C2^2xC4xC12");
// GroupNames label

G:=SmallGroup(192,1400);
// by ID

G=gap.SmallGroup(192,1400);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,680]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^4=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

׿
×
𝔽