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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), C4.11(C23×C6), C24.43(C2×C6), (C23×C4).17C6, (C23×C12).27C2, (C2×C6).385C24, (C2×C12).977C23, C22.14(C23×C6), C23.78(C22×C6), (C23×C6).120C22, (C22×C6).471C23, (C22×C12).605C22, (C22×C4).139(C2×C6), (C2×C4).145(C22×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

Generators and relations for Q8×C22×C6
 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 >

Subgroups: 850, all normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, Q8, C23, C12, C2×C6, C22×C4, C2×Q8, C24, C2×C12, C3×Q8, C22×C6, C23×C4, C22×Q8, C22×C12, C6×Q8, C23×C6, Q8×C23, C23×C12, Q8×C2×C6, Q8×C22×C6
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C24, C3×Q8, C22×C6, C22×Q8, C25, C6×Q8, C23×C6, Q8×C23, Q8×C2×C6, C24×C6, Q8×C22×C6

Smallest permutation representation of Q8×C22×C6
►Regular action on 192 points
Generators in S192
(1 54)(2 49)(3 50)(4 51)(5 52)(6 53)(7 156)(8 151)(9 152)(10 153)(11 154)(12 155)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(103 145)(104 146)(105 147)(106 148)(107 149)(108 150)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(157 187)(158 188)(159 189)(160 190)(161 191)(162 192)(163 181)(164 182)(165 183)(166 184)(167 185)(168 186)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)

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

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

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

Matrix representation of Q8×C22×C6 ►in 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] >;
 

Q8×C22×C6 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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