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G = Q8×C22×C6  order 192 = 26·3

Direct product of C22×C6 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C22×C6, C6.22C25, C12.88C24, C2.2(C24×C6), C24.43(C2×C6), C4.11(C23×C6), (C23×C4).17C6, (C23×C12).27C2, (C2×C6).385C24, (C2×C12).977C23, C22.14(C23×C6), C23.78(C22×C6), (C22×C6).471C23, (C23×C6).120C22, (C22×C12).605C22, (C2×C4).145(C22×C6), (C22×C4).139(C2×C6), SmallGroup(192,1532)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8×C22×C6
Chief series
C1 — C2 — C6 — C12 — C3×Q8 — C6×Q8 — Q8×C2×C6 — Q8×C22×C6
Lower central
C1 — C2 — Q8×C22×C6
Upper central
C1 — C23×C6 — Q8×C22×C6

Subgroups: 850, all normal (8 characteristic)
C1, C2, C2 [×14], C3, C4 [×24], C22 [×35], C6, C6 [×14], C2×C4 [×84], Q8 [×64], C23 [×15], C12 [×24], C2×C6 [×35], C22×C4 [×42], C2×Q8 [×112], C24, C2×C12 [×84], C3×Q8 [×64], C22×C6 [×15], C23×C4 [×3], C22×Q8 [×28], C22×C12 [×42], C6×Q8 [×112], C23×C6, Q8×C23, C23×C12 [×3], Q8×C2×C6 [×28], Q8×C22×C6

Quotients:
C1, C2 [×31], C3, C22 [×155], C6 [×31], Q8 [×8], C23 [×155], C2×C6 [×155], C2×Q8 [×28], C24 [×31], C3×Q8 [×8], C22×C6 [×155], C22×Q8 [×14], C25, C6×Q8 [×28], C23×C6 [×31], Q8×C23, Q8×C2×C6 [×14], C24×C6, Q8×C22×C6

Generators and relations
 G = < a,b,c,d,e | a2=b2=c6=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽13)

10000
012000
00100
000120
000012
,
120000
012000
00100
00010
00001
,
10000
01000
001200
00090
00009
,
120000
01000
001200
00001
000120
,
10000
01000
001200
00034
000410

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

120 conjugacy classes

class 1 2A···2O3A3B4A···4X6A···6AD12A···12AV
order12···2334···46···612···12
size11···1112···21···12···2

120 irreducible representations

dim11111122
type+++-
imageC1C2C2C3C6C6Q8C3×Q8
kernelQ8×C22×C6C23×C12Q8×C2×C6Q8×C23C23×C4C22×Q8C22×C6C23
# reps13282656816

In GAP, Magma, Sage, TeX 

Q_8\times C_2^2\times C_6
 
% in TeX
 
G:=Group("Q8xC2^2xC6");
 
// GroupNames label
 
G:=SmallGroup(192,1532);
 
// by ID
 
G=gap.SmallGroup(192,1532);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-2,672,1373,680]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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