direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C42.30C22, C4⋊Q8.8C6, C8⋊C4.6C6, (C2×C12).342D4, C42.28(C2×C6), Q8⋊C4.7C6, C42.C2.3C6, C22.112(C6×D4), C12.273(C4○D4), (C2×C12).947C23, (C4×C12).270C22, (C2×C24).337C22, C6.76(C4.4D4), (C6×Q8).175C22, C6.146(C8.C22), C4⋊C4.22(C2×C6), (C2×C8).58(C2×C6), C4.18(C3×C4○D4), (C2×C4).43(C3×D4), (C3×C4⋊Q8).23C2, (C2×C6).668(C2×D4), (C2×Q8).20(C2×C6), (C3×C8⋊C4).12C2, C2.14(C3×C4.4D4), C2.21(C3×C8.C22), (C3×C4⋊C4).242C22, (C2×C4).122(C22×C6), (C3×Q8⋊C4).16C2, (C3×C42.C2).10C2, SmallGroup(192,924)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.30C22
G = < a,b,c,d,e | a3=b4=c4=1, d2=c2, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, ebe-1=bc2, dcd-1=c-1, ce=ec, ede-1=b2c-1d >
Subgroups: 146 in 90 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, C42.30C22, C3×C8⋊C4, C3×Q8⋊C4, C3×C42.C2, C3×C4⋊Q8, C3×C42.30C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C8.C22, C6×D4, C3×C4○D4, C42.30C22, C3×C4.4D4, C3×C8.C22, C3×C42.30C22
►Regular action on 192 points
Generators in S192
(1 67 19)(2 68 20)(3 69 21)(4 70 22)(5 71 23)(6 72 24)(7 65 17)(8 66 18)(9 62 26)(10 63 27)(11 64 28)(12 57 29)(13 58 30)(14 59 31)(15 60 32)(16 61 25)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 101 89)(42 102 90)(43 103 91)(44 104 92)(45 97 93)(46 98 94)(47 99 95)(48 100 96)(49 190 138)(50 191 139)(51 192 140)(52 185 141)(53 186 142)(54 187 143)(55 188 144)(56 189 137)(105 149 157)(106 150 158)(107 151 159)(108 152 160)(109 145 153)(110 146 154)(111 147 155)(112 148 156)(113 121 165)(114 122 166)(115 123 167)(116 124 168)(117 125 161)(118 126 162)(119 127 163)(120 128 164)(129 173 181)(130 174 182)(131 175 183)(132 176 184)(133 169 177)(134 170 178)(135 171 179)(136 172 180)
(1 44 77 26)(2 41 78 31)(3 46 79 28)(4 43 80 25)(5 48 73 30)(6 45 74 27)(7 42 75 32)(8 47 76 29)(9 67 104 85)(10 72 97 82)(11 69 98 87)(12 66 99 84)(13 71 100 81)(14 68 101 86)(15 65 102 83)(16 70 103 88)(17 90 35 60)(18 95 36 57)(19 92 37 62)(20 89 38 59)(21 94 39 64)(22 91 40 61)(23 96 33 58)(24 93 34 63)(49 177 127 158)(50 182 128 155)(51 179 121 160)(52 184 122 157)(53 181 123 154)(54 178 124 159)(55 183 125 156)(56 180 126 153)(105 185 132 166)(106 190 133 163)(107 187 134 168)(108 192 135 165)(109 189 136 162)(110 186 129 167)(111 191 130 164)(112 188 131 161)(113 152 140 171)(114 149 141 176)(115 146 142 173)(116 151 143 170)(117 148 144 175)(118 145 137 172)(119 150 138 169)(120 147 139 174)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)(161 163 165 167)(162 164 166 168)(169 171 173 175)(170 172 174 176)(177 179 181 183)(178 180 182 184)(185 187 189 191)(186 188 190 192)
(1 171 5 175)(2 147 6 151)(3 169 7 173)(4 145 8 149)(9 121 13 125)(10 54 14 50)(11 127 15 123)(12 52 16 56)(17 129 21 133)(18 105 22 109)(19 135 23 131)(20 111 24 107)(25 137 29 141)(26 113 30 117)(27 143 31 139)(28 119 32 115)(33 112 37 108)(34 134 38 130)(35 110 39 106)(36 132 40 136)(41 120 45 116)(42 142 46 138)(43 118 47 114)(44 140 48 144)(49 102 53 98)(51 100 55 104)(57 185 61 189)(58 161 62 165)(59 191 63 187)(60 167 64 163)(65 181 69 177)(66 157 70 153)(67 179 71 183)(68 155 72 159)(73 148 77 152)(74 170 78 174)(75 146 79 150)(76 176 80 172)(81 156 85 160)(82 178 86 182)(83 154 87 158)(84 184 88 180)(89 164 93 168)(90 186 94 190)(91 162 95 166)(92 192 96 188)(97 124 101 128)(99 122 103 126)
(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)| (1,67,19)(2,68,20)(3,69,21)(4,70,22)(5,71,23)(6,72,24)(7,65,17)(8,66,18)(9,62,26)(10,63,27)(11,64,28)(12,57,29)(13,58,30)(14,59,31)(15,60,32)(16,61,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,101,89)(42,102,90)(43,103,91)(44,104,92)(45,97,93)(46,98,94)(47,99,95)(48,100,96)(49,190,138)(50,191,139)(51,192,140)(52,185,141)(53,186,142)(54,187,143)(55,188,144)(56,189,137)(105,149,157)(106,150,158)(107,151,159)(108,152,160)(109,145,153)(110,146,154)(111,147,155)(112,148,156)(113,121,165)(114,122,166)(115,123,167)(116,124,168)(117,125,161)(118,126,162)(119,127,163)(120,128,164)(129,173,181)(130,174,182)(131,175,183)(132,176,184)(133,169,177)(134,170,178)(135,171,179)(136,172,180), (1,44,77,26)(2,41,78,31)(3,46,79,28)(4,43,80,25)(5,48,73,30)(6,45,74,27)(7,42,75,32)(8,47,76,29)(9,67,104,85)(10,72,97,82)(11,69,98,87)(12,66,99,84)(13,71,100,81)(14,68,101,86)(15,65,102,83)(16,70,103,88)(17,90,35,60)(18,95,36,57)(19,92,37,62)(20,89,38,59)(21,94,39,64)(22,91,40,61)(23,96,33,58)(24,93,34,63)(49,177,127,158)(50,182,128,155)(51,179,121,160)(52,184,122,157)(53,181,123,154)(54,178,124,159)(55,183,125,156)(56,180,126,153)(105,185,132,166)(106,190,133,163)(107,187,134,168)(108,192,135,165)(109,189,136,162)(110,186,129,167)(111,191,130,164)(112,188,131,161)(113,152,140,171)(114,149,141,176)(115,146,142,173)(116,151,143,170)(117,148,144,175)(118,145,137,172)(119,150,138,169)(120,147,139,174), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192), (1,171,5,175)(2,147,6,151)(3,169,7,173)(4,145,8,149)(9,121,13,125)(10,54,14,50)(11,127,15,123)(12,52,16,56)(17,129,21,133)(18,105,22,109)(19,135,23,131)(20,111,24,107)(25,137,29,141)(26,113,30,117)(27,143,31,139)(28,119,32,115)(33,112,37,108)(34,134,38,130)(35,110,39,106)(36,132,40,136)(41,120,45,116)(42,142,46,138)(43,118,47,114)(44,140,48,144)(49,102,53,98)(51,100,55,104)(57,185,61,189)(58,161,62,165)(59,191,63,187)(60,167,64,163)(65,181,69,177)(66,157,70,153)(67,179,71,183)(68,155,72,159)(73,148,77,152)(74,170,78,174)(75,146,79,150)(76,176,80,172)(81,156,85,160)(82,178,86,182)(83,154,87,158)(84,184,88,180)(89,164,93,168)(90,186,94,190)(91,162,95,166)(92,192,96,188)(97,124,101,128)(99,122,103,126), (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( (1,67,19)(2,68,20)(3,69,21)(4,70,22)(5,71,23)(6,72,24)(7,65,17)(8,66,18)(9,62,26)(10,63,27)(11,64,28)(12,57,29)(13,58,30)(14,59,31)(15,60,32)(16,61,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,101,89)(42,102,90)(43,103,91)(44,104,92)(45,97,93)(46,98,94)(47,99,95)(48,100,96)(49,190,138)(50,191,139)(51,192,140)(52,185,141)(53,186,142)(54,187,143)(55,188,144)(56,189,137)(105,149,157)(106,150,158)(107,151,159)(108,152,160)(109,145,153)(110,146,154)(111,147,155)(112,148,156)(113,121,165)(114,122,166)(115,123,167)(116,124,168)(117,125,161)(118,126,162)(119,127,163)(120,128,164)(129,173,181)(130,174,182)(131,175,183)(132,176,184)(133,169,177)(134,170,178)(135,171,179)(136,172,180), (1,44,77,26)(2,41,78,31)(3,46,79,28)(4,43,80,25)(5,48,73,30)(6,45,74,27)(7,42,75,32)(8,47,76,29)(9,67,104,85)(10,72,97,82)(11,69,98,87)(12,66,99,84)(13,71,100,81)(14,68,101,86)(15,65,102,83)(16,70,103,88)(17,90,35,60)(18,95,36,57)(19,92,37,62)(20,89,38,59)(21,94,39,64)(22,91,40,61)(23,96,33,58)(24,93,34,63)(49,177,127,158)(50,182,128,155)(51,179,121,160)(52,184,122,157)(53,181,123,154)(54,178,124,159)(55,183,125,156)(56,180,126,153)(105,185,132,166)(106,190,133,163)(107,187,134,168)(108,192,135,165)(109,189,136,162)(110,186,129,167)(111,191,130,164)(112,188,131,161)(113,152,140,171)(114,149,141,176)(115,146,142,173)(116,151,143,170)(117,148,144,175)(118,145,137,172)(119,150,138,169)(120,147,139,174), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192), (1,171,5,175)(2,147,6,151)(3,169,7,173)(4,145,8,149)(9,121,13,125)(10,54,14,50)(11,127,15,123)(12,52,16,56)(17,129,21,133)(18,105,22,109)(19,135,23,131)(20,111,24,107)(25,137,29,141)(26,113,30,117)(27,143,31,139)(28,119,32,115)(33,112,37,108)(34,134,38,130)(35,110,39,106)(36,132,40,136)(41,120,45,116)(42,142,46,138)(43,118,47,114)(44,140,48,144)(49,102,53,98)(51,100,55,104)(57,185,61,189)(58,161,62,165)(59,191,63,187)(60,167,64,163)(65,181,69,177)(66,157,70,153)(67,179,71,183)(68,155,72,159)(73,148,77,152)(74,170,78,174)(75,146,79,150)(76,176,80,172)(81,156,85,160)(82,178,86,182)(83,154,87,158)(84,184,88,180)(89,164,93,168)(90,186,94,190)(91,162,95,166)(92,192,96,188)(97,124,101,128)(99,122,103,126), (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([[(1,67,19),(2,68,20),(3,69,21),(4,70,22),(5,71,23),(6,72,24),(7,65,17),(8,66,18),(9,62,26),(10,63,27),(11,64,28),(12,57,29),(13,58,30),(14,59,31),(15,60,32),(16,61,25),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,101,89),(42,102,90),(43,103,91),(44,104,92),(45,97,93),(46,98,94),(47,99,95),(48,100,96),(49,190,138),(50,191,139),(51,192,140),(52,185,141),(53,186,142),(54,187,143),(55,188,144),(56,189,137),(105,149,157),(106,150,158),(107,151,159),(108,152,160),(109,145,153),(110,146,154),(111,147,155),(112,148,156),(113,121,165),(114,122,166),(115,123,167),(116,124,168),(117,125,161),(118,126,162),(119,127,163),(120,128,164),(129,173,181),(130,174,182),(131,175,183),(132,176,184),(133,169,177),(134,170,178),(135,171,179),(136,172,180)], [(1,44,77,26),(2,41,78,31),(3,46,79,28),(4,43,80,25),(5,48,73,30),(6,45,74,27),(7,42,75,32),(8,47,76,29),(9,67,104,85),(10,72,97,82),(11,69,98,87),(12,66,99,84),(13,71,100,81),(14,68,101,86),(15,65,102,83),(16,70,103,88),(17,90,35,60),(18,95,36,57),(19,92,37,62),(20,89,38,59),(21,94,39,64),(22,91,40,61),(23,96,33,58),(24,93,34,63),(49,177,127,158),(50,182,128,155),(51,179,121,160),(52,184,122,157),(53,181,123,154),(54,178,124,159),(55,183,125,156),(56,180,126,153),(105,185,132,166),(106,190,133,163),(107,187,134,168),(108,192,135,165),(109,189,136,162),(110,186,129,167),(111,191,130,164),(112,188,131,161),(113,152,140,171),(114,149,141,176),(115,146,142,173),(116,151,143,170),(117,148,144,175),(118,145,137,172),(119,150,138,169),(120,147,139,174)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160),(161,163,165,167),(162,164,166,168),(169,171,173,175),(170,172,174,176),(177,179,181,183),(178,180,182,184),(185,187,189,191),(186,188,190,192)], [(1,171,5,175),(2,147,6,151),(3,169,7,173),(4,145,8,149),(9,121,13,125),(10,54,14,50),(11,127,15,123),(12,52,16,56),(17,129,21,133),(18,105,22,109),(19,135,23,131),(20,111,24,107),(25,137,29,141),(26,113,30,117),(27,143,31,139),(28,119,32,115),(33,112,37,108),(34,134,38,130),(35,110,39,106),(36,132,40,136),(41,120,45,116),(42,142,46,138),(43,118,47,114),(44,140,48,144),(49,102,53,98),(51,100,55,104),(57,185,61,189),(58,161,62,165),(59,191,63,187),(60,167,64,163),(65,181,69,177),(66,157,70,153),(67,179,71,183),(68,155,72,159),(73,148,77,152),(74,170,78,174),(75,146,79,150),(76,176,80,172),(81,156,85,160),(82,178,86,182),(83,154,87,158),(84,184,88,180),(89,164,93,168),(90,186,94,190),(91,162,95,166),(92,192,96,188),(97,124,101,128),(99,122,103,126)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 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 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | C8.C22 | C3×C8.C22 |
| kernel | C3×C42.30C22 | C3×C8⋊C4 | C3×Q8⋊C4 | C3×C42.C2 | C3×C4⋊Q8 | C42.30C22 | C8⋊C4 | Q8⋊C4 | C42.C2 | C4⋊Q8 | C2×C12 | C12 | C2×C4 | C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×C42.30C22 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 56 | 54 | 0 | 0 | 0 | 0 |
| 46 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 59 | 70 | 57 |
| 0 | 0 | 14 | 72 | 16 | 70 |
| 0 | 0 | 16 | 70 | 1 | 14 |
| 0 | 0 | 3 | 16 | 59 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 68 | 31 | 0 | 0 | 0 | 0 |
| 11 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 29 | 38 | 23 |
| 0 | 0 | 29 | 18 | 23 | 35 |
| 0 | 0 | 38 | 23 | 44 | 55 |
| 0 | 0 | 23 | 35 | 55 | 29 |
| 21 | 2 | 0 | 0 | 0 | 0 |
| 72 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 16 | 59 | 1 |
| 0 | 0 | 57 | 3 | 72 | 59 |
| 0 | 0 | 72 | 59 | 70 | 57 |
| 0 | 0 | 14 | 72 | 16 | 70 |
G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[56,46,0,0,0,0,54,17,0,0,0,0,0,0,72,14,16,3,0,0,59,72,70,16,0,0,70,16,1,59,0,0,57,70,14,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[68,11,0,0,0,0,31,5,0,0,0,0,0,0,55,29,38,23,0,0,29,18,23,35,0,0,38,23,44,55,0,0,23,35,55,29],[21,72,0,0,0,0,2,52,0,0,0,0,0,0,3,57,72,14,0,0,16,3,59,72,0,0,59,72,70,16,0,0,1,59,57,70] >;
C3×C42.30C22 in GAP, Magma, Sage, TeX
C_3\times C_4^2._{30}C_2^2 % in TeX
G:=Group("C3xC4^2.30C2^2"); // GroupNames label
G:=SmallGroup(192,924);
// by ID
G=gap.SmallGroup(192,924);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1016,1094,1059,142,4204,172,6053,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=1,d^2=c^2,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,e*b*e^-1=b*c^2,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=b^2*c^-1*d>;
// generators/relations