direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×Q8.Q8, C4⋊C8.8C6, Q8.2(C3×Q8), (C3×Q8).4Q8, C4.17(C6×Q8), C4.Q8.5C6, C2.D8.5C6, (C4×Q8).14C6, (C2×C12).335D4, C42.25(C2×C6), (Q8×C12).21C2, Q8⋊C4.5C6, C12.123(C2×Q8), C42.C2.2C6, C6.127(C4○D8), C22.100(C6×D4), C6.98(C22⋊Q8), C12.316(C4○D4), (C4×C12).267C22, (C2×C12).935C23, (C2×C24).189C22, (C6×Q8).265C22, C6.141(C8.C22), (C2×C8).8(C2×C6), (C3×C4⋊C8).18C2, C4⋊C4.16(C2×C6), C2.14(C3×C4○D8), C4.28(C3×C4○D4), (C2×C4).36(C3×D4), (C2×C6).656(C2×D4), (C2×Q8).64(C2×C6), (C3×C4.Q8).12C2, (C3×C2.D8).14C2, C2.17(C3×C22⋊Q8), C2.16(C3×C8.C22), (C3×C42.C2).9C2, (C3×C4⋊C4).379C22, (C2×C4).110(C22×C6), (C3×Q8⋊C4).14C2, SmallGroup(192,912)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8.Q8
G = < a,b,c,d,e | a3=b4=d4=1, c2=b2, e2=b2d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >
Subgroups: 138 in 90 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, Q8.Q8, C3×Q8⋊C4, C3×C4⋊C8, C3×C4.Q8, C3×C2.D8, Q8×C12, C3×C42.C2, C3×Q8.Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C4○D8, C8.C22, C6×D4, C6×Q8, C3×C4○D4, Q8.Q8, C3×C22⋊Q8, C3×C4○D8, C3×C8.C22, C3×Q8.Q8
►Regular action on 192 points
Generators in S192
(1 19 11)(2 20 12)(3 17 9)(4 18 10)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(97 113 105)(98 114 106)(99 115 107)(100 116 108)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(121 137 129)(122 138 130)(123 139 131)(124 140 132)(125 141 133)(126 142 134)(127 143 135)(128 144 136)(145 161 153)(146 162 154)(147 163 155)(148 164 156)(149 165 157)(150 166 158)(151 167 159)(152 168 160)(169 185 177)(170 186 178)(171 187 179)(172 188 180)(173 189 181)(174 190 182)(175 191 183)(176 192 184)
(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 27 3 25)(2 26 4 28)(5 29 7 31)(6 32 8 30)(9 33 11 35)(10 36 12 34)(13 37 15 39)(14 40 16 38)(17 41 19 43)(18 44 20 42)(21 45 23 47)(22 48 24 46)(49 73 51 75)(50 76 52 74)(53 77 55 79)(54 80 56 78)(57 81 59 83)(58 84 60 82)(61 85 63 87)(62 88 64 86)(65 89 67 91)(66 92 68 90)(69 93 71 95)(70 96 72 94)(97 124 99 122)(98 123 100 121)(101 128 103 126)(102 127 104 125)(105 132 107 130)(106 131 108 129)(109 136 111 134)(110 135 112 133)(113 140 115 138)(114 139 116 137)(117 144 119 142)(118 143 120 141)(145 172 147 170)(146 171 148 169)(149 176 151 174)(150 175 152 173)(153 180 155 178)(154 179 156 177)(157 184 159 182)(158 183 160 181)(161 188 163 186)(162 187 164 185)(165 192 167 190)(166 191 168 189)
(1 53 5 51)(2 54 6 52)(3 55 7 49)(4 56 8 50)(9 63 15 57)(10 64 16 58)(11 61 13 59)(12 62 14 60)(17 71 23 65)(18 72 24 66)(19 69 21 67)(20 70 22 68)(25 79 31 73)(26 80 32 74)(27 77 29 75)(28 78 30 76)(33 87 39 81)(34 88 40 82)(35 85 37 83)(36 86 38 84)(41 95 47 89)(42 96 48 90)(43 93 45 91)(44 94 46 92)(97 147 103 149)(98 148 104 150)(99 145 101 151)(100 146 102 152)(105 155 111 157)(106 156 112 158)(107 153 109 159)(108 154 110 160)(113 163 119 165)(114 164 120 166)(115 161 117 167)(116 162 118 168)(121 171 127 173)(122 172 128 174)(123 169 125 175)(124 170 126 176)(129 179 135 181)(130 180 136 182)(131 177 133 183)(132 178 134 184)(137 187 143 189)(138 188 144 190)(139 185 141 191)(140 186 142 192)
(1 101 7 97)(2 104 8 100)(3 103 5 99)(4 102 6 98)(9 111 13 107)(10 110 14 106)(11 109 15 105)(12 112 16 108)(17 119 21 115)(18 118 22 114)(19 117 23 113)(20 120 24 116)(25 127 29 123)(26 126 30 122)(27 125 31 121)(28 128 32 124)(33 135 37 131)(34 134 38 130)(35 133 39 129)(36 136 40 132)(41 143 45 139)(42 142 46 138)(43 141 47 137)(44 144 48 140)(49 151 53 147)(50 150 54 146)(51 149 55 145)(52 152 56 148)(57 159 61 155)(58 158 62 154)(59 157 63 153)(60 160 64 156)(65 167 69 163)(66 166 70 162)(67 165 71 161)(68 168 72 164)(73 175 77 171)(74 174 78 170)(75 173 79 169)(76 176 80 172)(81 183 85 179)(82 182 86 178)(83 181 87 177)(84 184 88 180)(89 191 93 187)(90 190 94 186)(91 189 95 185)(92 192 96 188)
G:=sub<Sym(192)| (1,19,11)(2,20,12)(3,17,9)(4,18,10)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136)(145,161,153)(146,162,154)(147,163,155)(148,164,156)(149,165,157)(150,166,158)(151,167,159)(152,168,160)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (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), (1,27,3,25)(2,26,4,28)(5,29,7,31)(6,32,8,30)(9,33,11,35)(10,36,12,34)(13,37,15,39)(14,40,16,38)(17,41,19,43)(18,44,20,42)(21,45,23,47)(22,48,24,46)(49,73,51,75)(50,76,52,74)(53,77,55,79)(54,80,56,78)(57,81,59,83)(58,84,60,82)(61,85,63,87)(62,88,64,86)(65,89,67,91)(66,92,68,90)(69,93,71,95)(70,96,72,94)(97,124,99,122)(98,123,100,121)(101,128,103,126)(102,127,104,125)(105,132,107,130)(106,131,108,129)(109,136,111,134)(110,135,112,133)(113,140,115,138)(114,139,116,137)(117,144,119,142)(118,143,120,141)(145,172,147,170)(146,171,148,169)(149,176,151,174)(150,175,152,173)(153,180,155,178)(154,179,156,177)(157,184,159,182)(158,183,160,181)(161,188,163,186)(162,187,164,185)(165,192,167,190)(166,191,168,189), (1,53,5,51)(2,54,6,52)(3,55,7,49)(4,56,8,50)(9,63,15,57)(10,64,16,58)(11,61,13,59)(12,62,14,60)(17,71,23,65)(18,72,24,66)(19,69,21,67)(20,70,22,68)(25,79,31,73)(26,80,32,74)(27,77,29,75)(28,78,30,76)(33,87,39,81)(34,88,40,82)(35,85,37,83)(36,86,38,84)(41,95,47,89)(42,96,48,90)(43,93,45,91)(44,94,46,92)(97,147,103,149)(98,148,104,150)(99,145,101,151)(100,146,102,152)(105,155,111,157)(106,156,112,158)(107,153,109,159)(108,154,110,160)(113,163,119,165)(114,164,120,166)(115,161,117,167)(116,162,118,168)(121,171,127,173)(122,172,128,174)(123,169,125,175)(124,170,126,176)(129,179,135,181)(130,180,136,182)(131,177,133,183)(132,178,134,184)(137,187,143,189)(138,188,144,190)(139,185,141,191)(140,186,142,192), (1,101,7,97)(2,104,8,100)(3,103,5,99)(4,102,6,98)(9,111,13,107)(10,110,14,106)(11,109,15,105)(12,112,16,108)(17,119,21,115)(18,118,22,114)(19,117,23,113)(20,120,24,116)(25,127,29,123)(26,126,30,122)(27,125,31,121)(28,128,32,124)(33,135,37,131)(34,134,38,130)(35,133,39,129)(36,136,40,132)(41,143,45,139)(42,142,46,138)(43,141,47,137)(44,144,48,140)(49,151,53,147)(50,150,54,146)(51,149,55,145)(52,152,56,148)(57,159,61,155)(58,158,62,154)(59,157,63,153)(60,160,64,156)(65,167,69,163)(66,166,70,162)(67,165,71,161)(68,168,72,164)(73,175,77,171)(74,174,78,170)(75,173,79,169)(76,176,80,172)(81,183,85,179)(82,182,86,178)(83,181,87,177)(84,184,88,180)(89,191,93,187)(90,190,94,186)(91,189,95,185)(92,192,96,188)>;
G:=Group( (1,19,11)(2,20,12)(3,17,9)(4,18,10)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136)(145,161,153)(146,162,154)(147,163,155)(148,164,156)(149,165,157)(150,166,158)(151,167,159)(152,168,160)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (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), (1,27,3,25)(2,26,4,28)(5,29,7,31)(6,32,8,30)(9,33,11,35)(10,36,12,34)(13,37,15,39)(14,40,16,38)(17,41,19,43)(18,44,20,42)(21,45,23,47)(22,48,24,46)(49,73,51,75)(50,76,52,74)(53,77,55,79)(54,80,56,78)(57,81,59,83)(58,84,60,82)(61,85,63,87)(62,88,64,86)(65,89,67,91)(66,92,68,90)(69,93,71,95)(70,96,72,94)(97,124,99,122)(98,123,100,121)(101,128,103,126)(102,127,104,125)(105,132,107,130)(106,131,108,129)(109,136,111,134)(110,135,112,133)(113,140,115,138)(114,139,116,137)(117,144,119,142)(118,143,120,141)(145,172,147,170)(146,171,148,169)(149,176,151,174)(150,175,152,173)(153,180,155,178)(154,179,156,177)(157,184,159,182)(158,183,160,181)(161,188,163,186)(162,187,164,185)(165,192,167,190)(166,191,168,189), (1,53,5,51)(2,54,6,52)(3,55,7,49)(4,56,8,50)(9,63,15,57)(10,64,16,58)(11,61,13,59)(12,62,14,60)(17,71,23,65)(18,72,24,66)(19,69,21,67)(20,70,22,68)(25,79,31,73)(26,80,32,74)(27,77,29,75)(28,78,30,76)(33,87,39,81)(34,88,40,82)(35,85,37,83)(36,86,38,84)(41,95,47,89)(42,96,48,90)(43,93,45,91)(44,94,46,92)(97,147,103,149)(98,148,104,150)(99,145,101,151)(100,146,102,152)(105,155,111,157)(106,156,112,158)(107,153,109,159)(108,154,110,160)(113,163,119,165)(114,164,120,166)(115,161,117,167)(116,162,118,168)(121,171,127,173)(122,172,128,174)(123,169,125,175)(124,170,126,176)(129,179,135,181)(130,180,136,182)(131,177,133,183)(132,178,134,184)(137,187,143,189)(138,188,144,190)(139,185,141,191)(140,186,142,192), (1,101,7,97)(2,104,8,100)(3,103,5,99)(4,102,6,98)(9,111,13,107)(10,110,14,106)(11,109,15,105)(12,112,16,108)(17,119,21,115)(18,118,22,114)(19,117,23,113)(20,120,24,116)(25,127,29,123)(26,126,30,122)(27,125,31,121)(28,128,32,124)(33,135,37,131)(34,134,38,130)(35,133,39,129)(36,136,40,132)(41,143,45,139)(42,142,46,138)(43,141,47,137)(44,144,48,140)(49,151,53,147)(50,150,54,146)(51,149,55,145)(52,152,56,148)(57,159,61,155)(58,158,62,154)(59,157,63,153)(60,160,64,156)(65,167,69,163)(66,166,70,162)(67,165,71,161)(68,168,72,164)(73,175,77,171)(74,174,78,170)(75,173,79,169)(76,176,80,172)(81,183,85,179)(82,182,86,178)(83,181,87,177)(84,184,88,180)(89,191,93,187)(90,190,94,186)(91,189,95,185)(92,192,96,188) );
G=PermutationGroup([[(1,19,11),(2,20,12),(3,17,9),(4,18,10),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,65,57),(50,66,58),(51,67,59),(52,68,60),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(97,113,105),(98,114,106),(99,115,107),(100,116,108),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(121,137,129),(122,138,130),(123,139,131),(124,140,132),(125,141,133),(126,142,134),(127,143,135),(128,144,136),(145,161,153),(146,162,154),(147,163,155),(148,164,156),(149,165,157),(150,166,158),(151,167,159),(152,168,160),(169,185,177),(170,186,178),(171,187,179),(172,188,180),(173,189,181),(174,190,182),(175,191,183),(176,192,184)], [(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)], [(1,27,3,25),(2,26,4,28),(5,29,7,31),(6,32,8,30),(9,33,11,35),(10,36,12,34),(13,37,15,39),(14,40,16,38),(17,41,19,43),(18,44,20,42),(21,45,23,47),(22,48,24,46),(49,73,51,75),(50,76,52,74),(53,77,55,79),(54,80,56,78),(57,81,59,83),(58,84,60,82),(61,85,63,87),(62,88,64,86),(65,89,67,91),(66,92,68,90),(69,93,71,95),(70,96,72,94),(97,124,99,122),(98,123,100,121),(101,128,103,126),(102,127,104,125),(105,132,107,130),(106,131,108,129),(109,136,111,134),(110,135,112,133),(113,140,115,138),(114,139,116,137),(117,144,119,142),(118,143,120,141),(145,172,147,170),(146,171,148,169),(149,176,151,174),(150,175,152,173),(153,180,155,178),(154,179,156,177),(157,184,159,182),(158,183,160,181),(161,188,163,186),(162,187,164,185),(165,192,167,190),(166,191,168,189)], [(1,53,5,51),(2,54,6,52),(3,55,7,49),(4,56,8,50),(9,63,15,57),(10,64,16,58),(11,61,13,59),(12,62,14,60),(17,71,23,65),(18,72,24,66),(19,69,21,67),(20,70,22,68),(25,79,31,73),(26,80,32,74),(27,77,29,75),(28,78,30,76),(33,87,39,81),(34,88,40,82),(35,85,37,83),(36,86,38,84),(41,95,47,89),(42,96,48,90),(43,93,45,91),(44,94,46,92),(97,147,103,149),(98,148,104,150),(99,145,101,151),(100,146,102,152),(105,155,111,157),(106,156,112,158),(107,153,109,159),(108,154,110,160),(113,163,119,165),(114,164,120,166),(115,161,117,167),(116,162,118,168),(121,171,127,173),(122,172,128,174),(123,169,125,175),(124,170,126,176),(129,179,135,181),(130,180,136,182),(131,177,133,183),(132,178,134,184),(137,187,143,189),(138,188,144,190),(139,185,141,191),(140,186,142,192)], [(1,101,7,97),(2,104,8,100),(3,103,5,99),(4,102,6,98),(9,111,13,107),(10,110,14,106),(11,109,15,105),(12,112,16,108),(17,119,21,115),(18,118,22,114),(19,117,23,113),(20,120,24,116),(25,127,29,123),(26,126,30,122),(27,125,31,121),(28,128,32,124),(33,135,37,131),(34,134,38,130),(35,133,39,129),(36,136,40,132),(41,143,45,139),(42,142,46,138),(43,141,47,137),(44,144,48,140),(49,151,53,147),(50,150,54,146),(51,149,55,145),(52,152,56,148),(57,159,61,155),(58,158,62,154),(59,157,63,153),(60,160,64,156),(65,167,69,163),(66,166,70,162),(67,165,71,161),(68,168,72,164),(73,175,77,171),(74,174,78,170),(75,173,79,169),(76,176,80,172),(81,183,85,179),(82,182,86,178),(83,181,87,177),(84,184,88,180),(89,191,93,187),(90,190,94,186),(91,189,95,185),(92,192,96,188)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12R | 12S | 12T | 12U | 12V | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | Q8 | C4○D4 | C3×D4 | C3×Q8 | C4○D8 | C3×C4○D4 | C3×C4○D8 | C8.C22 | C3×C8.C22 |
| kernel | C3×Q8.Q8 | C3×Q8⋊C4 | C3×C4⋊C8 | C3×C4.Q8 | C3×C2.D8 | Q8×C12 | C3×C42.C2 | Q8.Q8 | Q8⋊C4 | C4⋊C8 | C4.Q8 | C2.D8 | C4×Q8 | C42.C2 | C2×C12 | C3×Q8 | C12 | C2×C4 | Q8 | C6 | C4 | C2 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 |
Matrix representation of C3×Q8.Q8 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 54 | 21 |
| 0 | 0 | 21 | 19 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 61 | 52 |
| 0 | 0 | 52 | 12 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,54,21,0,0,21,19],[0,27,0,0,27,0,0,0,0,0,46,0,0,0,0,46],[0,1,0,0,72,0,0,0,0,0,61,52,0,0,52,12] >;
C3×Q8.Q8 in GAP, Magma, Sage, TeX
C_3\times Q_8.Q_8
% in TeX
G:=Group("C3xQ8.Q8"); // GroupNames label
G:=SmallGroup(192,912);
// by ID
G=gap.SmallGroup(192,912);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,848,1094,520,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^4=1,c^2=b^2,e^2=b^2*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d^-1>;
// generators/relations