(C2^2xC4).30D6 - GroupNames

G = (C₂²×C₄).₃₀D₆ order 192 = 2⁶·3

14^th non-split extension by C₂²×C₄ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂²×C₄).₃₀D₆, C₂.₅(C₄²⋊₃S₃), C₃⋊(C₂³.₈₄C₂³), C₂.C₄².₆S₃, C₆.C₄².₈C₂, C₆.₁₉(C₄²⋊₂C₂), C₂².₈₈(C₄○D₁₂), (C₂²×C₆).₂₈₇C₂³, (C₂²×C₁₂).₁₁C₂², C₂³.₃₆₈(C₂²×S₃), C₂².₈₇(D₄⋊₂S₃), C₂².₄₃(Q₈⋊₃S₃), C₂.₁₁(C₂³.₈D₆), (C₂²×Dic₃).₁₂C₂², C₂.₉(C₄⋊C₄⋊S₃), (C₂×C₆).₁₃₀(C₄○D₄), (C₃×C₂.C₄²).₂C₂, SmallGroup(192,221)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₆ — (C₂²×C₄).₃₀D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂²×Dic₃ — C₆.C₄² — (C₂²×C₄).₃₀D₆

Lower central

C₃ — C₂²×C₆ — (C₂²×C₄).₃₀D₆

Upper central

C₁ — C₂³ — C₂.C₄²

Generators and relations for (C₂²×C₄).₃₀D₆
G = < a,b,c,d,e | a²=b²=c⁴=1, d⁶=ba=ab, e²=bc², ac=ca, ad=da, ae=ea, dcd^-1=bc=cb, bd=db, be=eb, ece^-1=abc, ede^-1=abc²d⁵ >

Subgroups: 304 in 118 conjugacy classes, 51 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₂×C₄, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂²×C₄, C₂²×C₄, C₂×Dic₃, C₂×C₁₂, C₂²×C₆, C₂.C₄², C₂.C₄², C₂²×Dic₃, C₂²×Dic₃, C₂²×C₁₂, C₂³.₈₄C₂³, C₆.C₄², C₃×C₂.C₄², (C₂²×C₄).₃₀D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂²×S₃, C₄²⋊₂C₂, C₄○D₁₂, D₄⋊₂S₃, Q₈⋊₃S₃, C₂³.₈₄C₂³, C₄²⋊₃S₃, C₂³.₈D₆, C₄⋊C₄⋊S₃, (C₂²×C₄).₃₀D₆

Smallest permutation representation of (C₂²×C₄).₃₀D₆
►Regular action on 192 points

Generators in S₁₉₂

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

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

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

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

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

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

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

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	2	2	2	2	4	4
type	+	+	+	+	+			-	+
image	C₁	C₂	C₂	S₃	D₆	C₄○D₄	C₄○D₁₂	D₄⋊₂S₃	Q₈⋊₃S₃
kernel	(C₂²×C₄).₃₀D₆	C₆.C₄²	C₃×C₂.C₄²	C₂.C₄²	C₂²×C₄	C₂×C₆	C₂²	C₂²	C₂²
# reps	1	6	1	1	3	14	12	3	1

Matrix representation of (C₂²×C₄).₃₀D₆ ►in GL₆(𝔽₁₃)

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	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	1	0
0	0	0	0	0	1

12	0	0	0	0	0
8	1	0	0	0	0
0	0	4	2	0	0
0	0	11	9	0	0
0	0	0	0	10	6
0	0	0	0	7	3

5	11	0	0	0	0
12	8	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	8
0	0	0	0	5	8

8	0	0	0	0	0
0	8	0	0	0	0
0	0	7	10	0	0
0	0	3	6	0	0
0	0	0	0	11	4
0	0	0	0	2	2

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,8,0,0,0,0,0,1,0,0,0,0,0,0,4,11,0,0,0,0,2,9,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[5,12,0,0,0,0,11,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,8,8],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,7,3,0,0,0,0,10,6,0,0,0,0,0,0,11,2,0,0,0,0,4,2] >;

(C₂²×C₄).₃₀D₆ in GAP, Magma, Sage, TeX

(C_2^2\times C_4)._{30}D_6

% in TeX

G:=Group("(C2^2xC4).30D6");

// GroupNames label

G:=SmallGroup(192,221);

// by ID

G=gap.SmallGroup(192,221);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,64,1262,387,268,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^6=b*a=a*b,e^2=b*c^2,a*c=c*a,a*d=d*a,a*e=e*a,d*c*d^-1=b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=a*b*c,e*d*e^-1=a*b*c^2*d^5>;

// generators/relations

G = (C22×C4).30D6 order 192 = 26·3

14th non-split extension by C22×C4 of D6 acting via D6/C3=C22

G = (C₂²×C₄).₃₀D₆ order 192 = 2⁶·3

14^th non-split extension by C₂²×C₄ of D₆ acting via D₆/C₃=C₂²