metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.27(C4⋊Q8), (C2×C12).287D4, (C2×Dic3).7Q8, (C22×C4).57D6, C2.8(D6⋊3D4), C22.48(S3×Q8), C6.90(C4⋊D4), (C2×Dic3).60D4, C22.246(S3×D4), C6.47(C22⋊Q8), C2.21(D6⋊Q8), C6.22(C42.C2), C2.7(Dic3⋊Q8), C2.13(Dic3.Q8), C6.C42.19C2, C23.387(C22×S3), (C22×C6).347C23, (C22×C12).29C22, C22.105(C4○D12), C3⋊5(C23.81C23), C6.62(C22.D4), C22.101(D4⋊2S3), C2.12(C23.28D6), (C22×Dic3).55C22, (C6×C4⋊C4).31C2, (C2×C4⋊C4).20S3, (C2×C6).81(C2×Q8), (C2×C6).448(C2×D4), (C2×C4).38(C3⋊D4), (C2×C6).154(C4○D4), (C2×Dic3⋊C4).15C2, C22.137(C2×C3⋊D4), SmallGroup(192,542)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×Dic3).Q8
G = < a,b,c,d,e | a2=b6=d4=1, c2=b3, e2=ad2, ab=ba, ece-1=ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=b3c, ede-1=d-1 >
Subgroups: 360 in 150 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C4⋊C4, C2×C4⋊C4, Dic3⋊C4, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23.81C23, C6.C42, C6.C42, C2×Dic3⋊C4, C2×Dic3⋊C4, C6×C4⋊C4, (C2×Dic3).Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C4○D12, S3×D4, D4⋊2S3, S3×Q8, C2×C3⋊D4, C23.81C23, Dic3.Q8, D6⋊Q8, C23.28D6, D6⋊3D4, Dic3⋊Q8, (C2×Dic3).Q8
►Regular action on 192 points
Generators in S192
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 185)(14 186)(15 181)(16 182)(17 183)(18 184)(31 45)(32 46)(33 47)(34 48)(35 43)(36 44)(37 53)(38 54)(39 49)(40 50)(41 51)(42 52)(55 69)(56 70)(57 71)(58 72)(59 67)(60 68)(61 77)(62 78)(63 73)(64 74)(65 75)(66 76)(79 93)(80 94)(81 95)(82 96)(83 91)(84 92)(85 102)(86 97)(87 98)(88 99)(89 100)(90 101)(103 116)(104 117)(105 118)(106 119)(107 120)(108 115)(109 126)(110 121)(111 122)(112 123)(113 124)(114 125)(127 140)(128 141)(129 142)(130 143)(131 144)(132 139)(133 150)(134 145)(135 146)(136 147)(137 148)(138 149)(151 164)(152 165)(153 166)(154 167)(155 168)(156 163)(157 174)(158 169)(159 170)(160 171)(161 172)(162 173)(175 188)(176 189)(177 190)(178 191)(179 192)(180 187)
(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)
(1 188 4 191)(2 187 5 190)(3 192 6 189)(7 186 10 183)(8 185 11 182)(9 184 12 181)(13 29 16 26)(14 28 17 25)(15 27 18 30)(19 177 22 180)(20 176 23 179)(21 175 24 178)(31 159 34 162)(32 158 35 161)(33 157 36 160)(37 154 40 151)(38 153 41 156)(39 152 42 155)(43 172 46 169)(44 171 47 174)(45 170 48 173)(49 165 52 168)(50 164 53 167)(51 163 54 166)(55 131 58 128)(56 130 59 127)(57 129 60 132)(61 134 64 137)(62 133 65 136)(63 138 66 135)(67 140 70 143)(68 139 71 142)(69 144 72 141)(73 149 76 146)(74 148 77 145)(75 147 78 150)(79 111 82 114)(80 110 83 113)(81 109 84 112)(85 106 88 103)(86 105 89 108)(87 104 90 107)(91 124 94 121)(92 123 95 126)(93 122 96 125)(97 118 100 115)(98 117 101 120)(99 116 102 119)
(1 167 26 169)(2 166 27 174)(3 165 28 173)(4 164 29 172)(5 163 30 171)(6 168 25 170)(7 159 20 155)(8 158 21 154)(9 157 22 153)(10 162 23 152)(11 161 24 151)(12 160 19 156)(13 46 188 53)(14 45 189 52)(15 44 190 51)(16 43 191 50)(17 48 192 49)(18 47 187 54)(31 176 42 186)(32 175 37 185)(33 180 38 184)(34 179 39 183)(35 178 40 182)(36 177 41 181)(55 125 66 117)(56 124 61 116)(57 123 62 115)(58 122 63 120)(59 121 64 119)(60 126 65 118)(67 110 74 106)(68 109 75 105)(69 114 76 104)(70 113 77 103)(71 112 78 108)(72 111 73 107)(79 149 90 141)(80 148 85 140)(81 147 86 139)(82 146 87 144)(83 145 88 143)(84 150 89 142)(91 134 99 130)(92 133 100 129)(93 138 101 128)(94 137 102 127)(95 136 97 132)(96 135 98 131)
(1 74 8 59)(2 75 9 60)(3 76 10 55)(4 77 11 56)(5 78 12 57)(6 73 7 58)(13 127 175 148)(14 128 176 149)(15 129 177 150)(16 130 178 145)(17 131 179 146)(18 132 180 147)(19 62 30 71)(20 63 25 72)(21 64 26 67)(22 65 27 68)(23 66 28 69)(24 61 29 70)(31 90 52 93)(32 85 53 94)(33 86 54 95)(34 87 49 96)(35 88 50 91)(36 89 51 92)(37 80 46 102)(38 81 47 97)(39 82 48 98)(40 83 43 99)(41 84 44 100)(42 79 45 101)(103 151 124 172)(104 152 125 173)(105 153 126 174)(106 154 121 169)(107 155 122 170)(108 156 123 171)(109 157 118 166)(110 158 119 167)(111 159 120 168)(112 160 115 163)(113 161 116 164)(114 162 117 165)(133 181 142 190)(134 182 143 191)(135 183 144 192)(136 184 139 187)(137 185 140 188)(138 186 141 189)
G:=sub<Sym(192)| (1,21)(2,22)(3,23)(4,24)(5,19)(6,20)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,185)(14,186)(15,181)(16,182)(17,183)(18,184)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44)(37,53)(38,54)(39,49)(40,50)(41,51)(42,52)(55,69)(56,70)(57,71)(58,72)(59,67)(60,68)(61,77)(62,78)(63,73)(64,74)(65,75)(66,76)(79,93)(80,94)(81,95)(82,96)(83,91)(84,92)(85,102)(86,97)(87,98)(88,99)(89,100)(90,101)(103,116)(104,117)(105,118)(106,119)(107,120)(108,115)(109,126)(110,121)(111,122)(112,123)(113,124)(114,125)(127,140)(128,141)(129,142)(130,143)(131,144)(132,139)(133,150)(134,145)(135,146)(136,147)(137,148)(138,149)(151,164)(152,165)(153,166)(154,167)(155,168)(156,163)(157,174)(158,169)(159,170)(160,171)(161,172)(162,173)(175,188)(176,189)(177,190)(178,191)(179,192)(180,187), (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), (1,188,4,191)(2,187,5,190)(3,192,6,189)(7,186,10,183)(8,185,11,182)(9,184,12,181)(13,29,16,26)(14,28,17,25)(15,27,18,30)(19,177,22,180)(20,176,23,179)(21,175,24,178)(31,159,34,162)(32,158,35,161)(33,157,36,160)(37,154,40,151)(38,153,41,156)(39,152,42,155)(43,172,46,169)(44,171,47,174)(45,170,48,173)(49,165,52,168)(50,164,53,167)(51,163,54,166)(55,131,58,128)(56,130,59,127)(57,129,60,132)(61,134,64,137)(62,133,65,136)(63,138,66,135)(67,140,70,143)(68,139,71,142)(69,144,72,141)(73,149,76,146)(74,148,77,145)(75,147,78,150)(79,111,82,114)(80,110,83,113)(81,109,84,112)(85,106,88,103)(86,105,89,108)(87,104,90,107)(91,124,94,121)(92,123,95,126)(93,122,96,125)(97,118,100,115)(98,117,101,120)(99,116,102,119), (1,167,26,169)(2,166,27,174)(3,165,28,173)(4,164,29,172)(5,163,30,171)(6,168,25,170)(7,159,20,155)(8,158,21,154)(9,157,22,153)(10,162,23,152)(11,161,24,151)(12,160,19,156)(13,46,188,53)(14,45,189,52)(15,44,190,51)(16,43,191,50)(17,48,192,49)(18,47,187,54)(31,176,42,186)(32,175,37,185)(33,180,38,184)(34,179,39,183)(35,178,40,182)(36,177,41,181)(55,125,66,117)(56,124,61,116)(57,123,62,115)(58,122,63,120)(59,121,64,119)(60,126,65,118)(67,110,74,106)(68,109,75,105)(69,114,76,104)(70,113,77,103)(71,112,78,108)(72,111,73,107)(79,149,90,141)(80,148,85,140)(81,147,86,139)(82,146,87,144)(83,145,88,143)(84,150,89,142)(91,134,99,130)(92,133,100,129)(93,138,101,128)(94,137,102,127)(95,136,97,132)(96,135,98,131), (1,74,8,59)(2,75,9,60)(3,76,10,55)(4,77,11,56)(5,78,12,57)(6,73,7,58)(13,127,175,148)(14,128,176,149)(15,129,177,150)(16,130,178,145)(17,131,179,146)(18,132,180,147)(19,62,30,71)(20,63,25,72)(21,64,26,67)(22,65,27,68)(23,66,28,69)(24,61,29,70)(31,90,52,93)(32,85,53,94)(33,86,54,95)(34,87,49,96)(35,88,50,91)(36,89,51,92)(37,80,46,102)(38,81,47,97)(39,82,48,98)(40,83,43,99)(41,84,44,100)(42,79,45,101)(103,151,124,172)(104,152,125,173)(105,153,126,174)(106,154,121,169)(107,155,122,170)(108,156,123,171)(109,157,118,166)(110,158,119,167)(111,159,120,168)(112,160,115,163)(113,161,116,164)(114,162,117,165)(133,181,142,190)(134,182,143,191)(135,183,144,192)(136,184,139,187)(137,185,140,188)(138,186,141,189)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,19)(6,20)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,185)(14,186)(15,181)(16,182)(17,183)(18,184)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44)(37,53)(38,54)(39,49)(40,50)(41,51)(42,52)(55,69)(56,70)(57,71)(58,72)(59,67)(60,68)(61,77)(62,78)(63,73)(64,74)(65,75)(66,76)(79,93)(80,94)(81,95)(82,96)(83,91)(84,92)(85,102)(86,97)(87,98)(88,99)(89,100)(90,101)(103,116)(104,117)(105,118)(106,119)(107,120)(108,115)(109,126)(110,121)(111,122)(112,123)(113,124)(114,125)(127,140)(128,141)(129,142)(130,143)(131,144)(132,139)(133,150)(134,145)(135,146)(136,147)(137,148)(138,149)(151,164)(152,165)(153,166)(154,167)(155,168)(156,163)(157,174)(158,169)(159,170)(160,171)(161,172)(162,173)(175,188)(176,189)(177,190)(178,191)(179,192)(180,187), (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), (1,188,4,191)(2,187,5,190)(3,192,6,189)(7,186,10,183)(8,185,11,182)(9,184,12,181)(13,29,16,26)(14,28,17,25)(15,27,18,30)(19,177,22,180)(20,176,23,179)(21,175,24,178)(31,159,34,162)(32,158,35,161)(33,157,36,160)(37,154,40,151)(38,153,41,156)(39,152,42,155)(43,172,46,169)(44,171,47,174)(45,170,48,173)(49,165,52,168)(50,164,53,167)(51,163,54,166)(55,131,58,128)(56,130,59,127)(57,129,60,132)(61,134,64,137)(62,133,65,136)(63,138,66,135)(67,140,70,143)(68,139,71,142)(69,144,72,141)(73,149,76,146)(74,148,77,145)(75,147,78,150)(79,111,82,114)(80,110,83,113)(81,109,84,112)(85,106,88,103)(86,105,89,108)(87,104,90,107)(91,124,94,121)(92,123,95,126)(93,122,96,125)(97,118,100,115)(98,117,101,120)(99,116,102,119), (1,167,26,169)(2,166,27,174)(3,165,28,173)(4,164,29,172)(5,163,30,171)(6,168,25,170)(7,159,20,155)(8,158,21,154)(9,157,22,153)(10,162,23,152)(11,161,24,151)(12,160,19,156)(13,46,188,53)(14,45,189,52)(15,44,190,51)(16,43,191,50)(17,48,192,49)(18,47,187,54)(31,176,42,186)(32,175,37,185)(33,180,38,184)(34,179,39,183)(35,178,40,182)(36,177,41,181)(55,125,66,117)(56,124,61,116)(57,123,62,115)(58,122,63,120)(59,121,64,119)(60,126,65,118)(67,110,74,106)(68,109,75,105)(69,114,76,104)(70,113,77,103)(71,112,78,108)(72,111,73,107)(79,149,90,141)(80,148,85,140)(81,147,86,139)(82,146,87,144)(83,145,88,143)(84,150,89,142)(91,134,99,130)(92,133,100,129)(93,138,101,128)(94,137,102,127)(95,136,97,132)(96,135,98,131), (1,74,8,59)(2,75,9,60)(3,76,10,55)(4,77,11,56)(5,78,12,57)(6,73,7,58)(13,127,175,148)(14,128,176,149)(15,129,177,150)(16,130,178,145)(17,131,179,146)(18,132,180,147)(19,62,30,71)(20,63,25,72)(21,64,26,67)(22,65,27,68)(23,66,28,69)(24,61,29,70)(31,90,52,93)(32,85,53,94)(33,86,54,95)(34,87,49,96)(35,88,50,91)(36,89,51,92)(37,80,46,102)(38,81,47,97)(39,82,48,98)(40,83,43,99)(41,84,44,100)(42,79,45,101)(103,151,124,172)(104,152,125,173)(105,153,126,174)(106,154,121,169)(107,155,122,170)(108,156,123,171)(109,157,118,166)(110,158,119,167)(111,159,120,168)(112,160,115,163)(113,161,116,164)(114,162,117,165)(133,181,142,190)(134,182,143,191)(135,183,144,192)(136,184,139,187)(137,185,140,188)(138,186,141,189) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,19),(6,20),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,185),(14,186),(15,181),(16,182),(17,183),(18,184),(31,45),(32,46),(33,47),(34,48),(35,43),(36,44),(37,53),(38,54),(39,49),(40,50),(41,51),(42,52),(55,69),(56,70),(57,71),(58,72),(59,67),(60,68),(61,77),(62,78),(63,73),(64,74),(65,75),(66,76),(79,93),(80,94),(81,95),(82,96),(83,91),(84,92),(85,102),(86,97),(87,98),(88,99),(89,100),(90,101),(103,116),(104,117),(105,118),(106,119),(107,120),(108,115),(109,126),(110,121),(111,122),(112,123),(113,124),(114,125),(127,140),(128,141),(129,142),(130,143),(131,144),(132,139),(133,150),(134,145),(135,146),(136,147),(137,148),(138,149),(151,164),(152,165),(153,166),(154,167),(155,168),(156,163),(157,174),(158,169),(159,170),(160,171),(161,172),(162,173),(175,188),(176,189),(177,190),(178,191),(179,192),(180,187)], [(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)], [(1,188,4,191),(2,187,5,190),(3,192,6,189),(7,186,10,183),(8,185,11,182),(9,184,12,181),(13,29,16,26),(14,28,17,25),(15,27,18,30),(19,177,22,180),(20,176,23,179),(21,175,24,178),(31,159,34,162),(32,158,35,161),(33,157,36,160),(37,154,40,151),(38,153,41,156),(39,152,42,155),(43,172,46,169),(44,171,47,174),(45,170,48,173),(49,165,52,168),(50,164,53,167),(51,163,54,166),(55,131,58,128),(56,130,59,127),(57,129,60,132),(61,134,64,137),(62,133,65,136),(63,138,66,135),(67,140,70,143),(68,139,71,142),(69,144,72,141),(73,149,76,146),(74,148,77,145),(75,147,78,150),(79,111,82,114),(80,110,83,113),(81,109,84,112),(85,106,88,103),(86,105,89,108),(87,104,90,107),(91,124,94,121),(92,123,95,126),(93,122,96,125),(97,118,100,115),(98,117,101,120),(99,116,102,119)], [(1,167,26,169),(2,166,27,174),(3,165,28,173),(4,164,29,172),(5,163,30,171),(6,168,25,170),(7,159,20,155),(8,158,21,154),(9,157,22,153),(10,162,23,152),(11,161,24,151),(12,160,19,156),(13,46,188,53),(14,45,189,52),(15,44,190,51),(16,43,191,50),(17,48,192,49),(18,47,187,54),(31,176,42,186),(32,175,37,185),(33,180,38,184),(34,179,39,183),(35,178,40,182),(36,177,41,181),(55,125,66,117),(56,124,61,116),(57,123,62,115),(58,122,63,120),(59,121,64,119),(60,126,65,118),(67,110,74,106),(68,109,75,105),(69,114,76,104),(70,113,77,103),(71,112,78,108),(72,111,73,107),(79,149,90,141),(80,148,85,140),(81,147,86,139),(82,146,87,144),(83,145,88,143),(84,150,89,142),(91,134,99,130),(92,133,100,129),(93,138,101,128),(94,137,102,127),(95,136,97,132),(96,135,98,131)], [(1,74,8,59),(2,75,9,60),(3,76,10,55),(4,77,11,56),(5,78,12,57),(6,73,7,58),(13,127,175,148),(14,128,176,149),(15,129,177,150),(16,130,178,145),(17,131,179,146),(18,132,180,147),(19,62,30,71),(20,63,25,72),(21,64,26,67),(22,65,27,68),(23,66,28,69),(24,61,29,70),(31,90,52,93),(32,85,53,94),(33,86,54,95),(34,87,49,96),(35,88,50,91),(36,89,51,92),(37,80,46,102),(38,81,47,97),(39,82,48,98),(40,83,43,99),(41,84,44,100),(42,79,45,101),(103,151,124,172),(104,152,125,173),(105,153,126,174),(106,154,121,169),(107,155,122,170),(108,156,123,171),(109,157,118,166),(110,158,119,167),(111,159,120,168),(112,160,115,163),(113,161,116,164),(114,162,117,165),(133,181,142,190),(134,182,143,191),(135,183,144,192),(136,184,139,187),(137,185,140,188),(138,186,141,189)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4F | 4G | ··· | 4N | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | Q8 | D4 | D6 | C4○D4 | C3⋊D4 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 |
| kernel | (C2×Dic3).Q8 | C6.C42 | C2×Dic3⋊C4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 3 | 1 | 1 | 2 | 4 | 2 | 3 | 6 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of (C2×Dic3).Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,10,7,0,0,0,0,10,3],[1,0,0,0,0,0,2,12,0,0,0,0,0,0,8,0,0,0,0,0,3,5,0,0,0,0,0,0,2,4,0,0,0,0,2,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,11,9,0,0,0,0,4,2] >;
(C2×Dic3).Q8 in GAP, Magma, Sage, TeX
(C_2\times {\rm Dic}_3).Q_8 % in TeX
G:=Group("(C2xDic3).Q8"); // GroupNames label
G:=SmallGroup(192,542);
// by ID
G=gap.SmallGroup(192,542);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,254,387,100,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=d^4=1,c^2=b^3,e^2=a*d^2,a*b=b*a,e*c*e^-1=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations