direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C16⋊5C4, C48⋊11C4, C16⋊5C12, C24.8C8, C8.2C24, C42.4C12, C6.7M5(2), C12.34C42, (C4×C8).14C6, (C4×C12).7C4, C2.3(C4×C24), (C2×C4).2C24, (C2×C12).6C8, (C2×C16).7C6, C4.6(C4×C12), C6.12(C4×C8), (C4×C24).32C2, (C2×C48).17C2, (C2×C24).31C4, C4.12(C2×C24), (C2×C8).12C12, C8.21(C2×C12), C12.52(C2×C8), C24.90(C2×C4), C22.8(C2×C24), C2.1(C3×M5(2)), (C2×C24).450C22, (C2×C6).39(C2×C8), (C2×C4).82(C2×C12), (C2×C8).104(C2×C6), (C2×C12).344(C2×C4), SmallGroup(192,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C16⋊5C4
G = < a,b,c | a3=b16=c4=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C3×C16⋊5C4
►Regular action on 192 points
Generators in S192
(1 119 31)(2 120 32)(3 121 17)(4 122 18)(5 123 19)(6 124 20)(7 125 21)(8 126 22)(9 127 23)(10 128 24)(11 113 25)(12 114 26)(13 115 27)(14 116 28)(15 117 29)(16 118 30)(33 50 175)(34 51 176)(35 52 161)(36 53 162)(37 54 163)(38 55 164)(39 56 165)(40 57 166)(41 58 167)(42 59 168)(43 60 169)(44 61 170)(45 62 171)(46 63 172)(47 64 173)(48 49 174)(65 147 136)(66 148 137)(67 149 138)(68 150 139)(69 151 140)(70 152 141)(71 153 142)(72 154 143)(73 155 144)(74 156 129)(75 157 130)(76 158 131)(77 159 132)(78 160 133)(79 145 134)(80 146 135)(81 192 105)(82 177 106)(83 178 107)(84 179 108)(85 180 109)(86 181 110)(87 182 111)(88 183 112)(89 184 97)(90 185 98)(91 186 99)(92 187 100)(93 188 101)(94 189 102)(95 190 103)(96 191 104)
(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)
(1 164 72 179)(2 173 73 188)(3 166 74 181)(4 175 75 190)(5 168 76 183)(6 161 77 192)(7 170 78 185)(8 163 79 178)(9 172 80 187)(10 165 65 180)(11 174 66 189)(12 167 67 182)(13 176 68 191)(14 169 69 184)(15 162 70 177)(16 171 71 186)(17 57 129 86)(18 50 130 95)(19 59 131 88)(20 52 132 81)(21 61 133 90)(22 54 134 83)(23 63 135 92)(24 56 136 85)(25 49 137 94)(26 58 138 87)(27 51 139 96)(28 60 140 89)(29 53 141 82)(30 62 142 91)(31 55 143 84)(32 64 144 93)(33 157 103 122)(34 150 104 115)(35 159 105 124)(36 152 106 117)(37 145 107 126)(38 154 108 119)(39 147 109 128)(40 156 110 121)(41 149 111 114)(42 158 112 123)(43 151 97 116)(44 160 98 125)(45 153 99 118)(46 146 100 127)(47 155 101 120)(48 148 102 113)
G:=sub<Sym(192)| (1,119,31)(2,120,32)(3,121,17)(4,122,18)(5,123,19)(6,124,20)(7,125,21)(8,126,22)(9,127,23)(10,128,24)(11,113,25)(12,114,26)(13,115,27)(14,116,28)(15,117,29)(16,118,30)(33,50,175)(34,51,176)(35,52,161)(36,53,162)(37,54,163)(38,55,164)(39,56,165)(40,57,166)(41,58,167)(42,59,168)(43,60,169)(44,61,170)(45,62,171)(46,63,172)(47,64,173)(48,49,174)(65,147,136)(66,148,137)(67,149,138)(68,150,139)(69,151,140)(70,152,141)(71,153,142)(72,154,143)(73,155,144)(74,156,129)(75,157,130)(76,158,131)(77,159,132)(78,160,133)(79,145,134)(80,146,135)(81,192,105)(82,177,106)(83,178,107)(84,179,108)(85,180,109)(86,181,110)(87,182,111)(88,183,112)(89,184,97)(90,185,98)(91,186,99)(92,187,100)(93,188,101)(94,189,102)(95,190,103)(96,191,104), (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), (1,164,72,179)(2,173,73,188)(3,166,74,181)(4,175,75,190)(5,168,76,183)(6,161,77,192)(7,170,78,185)(8,163,79,178)(9,172,80,187)(10,165,65,180)(11,174,66,189)(12,167,67,182)(13,176,68,191)(14,169,69,184)(15,162,70,177)(16,171,71,186)(17,57,129,86)(18,50,130,95)(19,59,131,88)(20,52,132,81)(21,61,133,90)(22,54,134,83)(23,63,135,92)(24,56,136,85)(25,49,137,94)(26,58,138,87)(27,51,139,96)(28,60,140,89)(29,53,141,82)(30,62,142,91)(31,55,143,84)(32,64,144,93)(33,157,103,122)(34,150,104,115)(35,159,105,124)(36,152,106,117)(37,145,107,126)(38,154,108,119)(39,147,109,128)(40,156,110,121)(41,149,111,114)(42,158,112,123)(43,151,97,116)(44,160,98,125)(45,153,99,118)(46,146,100,127)(47,155,101,120)(48,148,102,113)>;
G:=Group( (1,119,31)(2,120,32)(3,121,17)(4,122,18)(5,123,19)(6,124,20)(7,125,21)(8,126,22)(9,127,23)(10,128,24)(11,113,25)(12,114,26)(13,115,27)(14,116,28)(15,117,29)(16,118,30)(33,50,175)(34,51,176)(35,52,161)(36,53,162)(37,54,163)(38,55,164)(39,56,165)(40,57,166)(41,58,167)(42,59,168)(43,60,169)(44,61,170)(45,62,171)(46,63,172)(47,64,173)(48,49,174)(65,147,136)(66,148,137)(67,149,138)(68,150,139)(69,151,140)(70,152,141)(71,153,142)(72,154,143)(73,155,144)(74,156,129)(75,157,130)(76,158,131)(77,159,132)(78,160,133)(79,145,134)(80,146,135)(81,192,105)(82,177,106)(83,178,107)(84,179,108)(85,180,109)(86,181,110)(87,182,111)(88,183,112)(89,184,97)(90,185,98)(91,186,99)(92,187,100)(93,188,101)(94,189,102)(95,190,103)(96,191,104), (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), (1,164,72,179)(2,173,73,188)(3,166,74,181)(4,175,75,190)(5,168,76,183)(6,161,77,192)(7,170,78,185)(8,163,79,178)(9,172,80,187)(10,165,65,180)(11,174,66,189)(12,167,67,182)(13,176,68,191)(14,169,69,184)(15,162,70,177)(16,171,71,186)(17,57,129,86)(18,50,130,95)(19,59,131,88)(20,52,132,81)(21,61,133,90)(22,54,134,83)(23,63,135,92)(24,56,136,85)(25,49,137,94)(26,58,138,87)(27,51,139,96)(28,60,140,89)(29,53,141,82)(30,62,142,91)(31,55,143,84)(32,64,144,93)(33,157,103,122)(34,150,104,115)(35,159,105,124)(36,152,106,117)(37,145,107,126)(38,154,108,119)(39,147,109,128)(40,156,110,121)(41,149,111,114)(42,158,112,123)(43,151,97,116)(44,160,98,125)(45,153,99,118)(46,146,100,127)(47,155,101,120)(48,148,102,113) );
G=PermutationGroup([[(1,119,31),(2,120,32),(3,121,17),(4,122,18),(5,123,19),(6,124,20),(7,125,21),(8,126,22),(9,127,23),(10,128,24),(11,113,25),(12,114,26),(13,115,27),(14,116,28),(15,117,29),(16,118,30),(33,50,175),(34,51,176),(35,52,161),(36,53,162),(37,54,163),(38,55,164),(39,56,165),(40,57,166),(41,58,167),(42,59,168),(43,60,169),(44,61,170),(45,62,171),(46,63,172),(47,64,173),(48,49,174),(65,147,136),(66,148,137),(67,149,138),(68,150,139),(69,151,140),(70,152,141),(71,153,142),(72,154,143),(73,155,144),(74,156,129),(75,157,130),(76,158,131),(77,159,132),(78,160,133),(79,145,134),(80,146,135),(81,192,105),(82,177,106),(83,178,107),(84,179,108),(85,180,109),(86,181,110),(87,182,111),(88,183,112),(89,184,97),(90,185,98),(91,186,99),(92,187,100),(93,188,101),(94,189,102),(95,190,103),(96,191,104)], [(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)], [(1,164,72,179),(2,173,73,188),(3,166,74,181),(4,175,75,190),(5,168,76,183),(6,161,77,192),(7,170,78,185),(8,163,79,178),(9,172,80,187),(10,165,65,180),(11,174,66,189),(12,167,67,182),(13,176,68,191),(14,169,69,184),(15,162,70,177),(16,171,71,186),(17,57,129,86),(18,50,130,95),(19,59,131,88),(20,52,132,81),(21,61,133,90),(22,54,134,83),(23,63,135,92),(24,56,136,85),(25,49,137,94),(26,58,138,87),(27,51,139,96),(28,60,140,89),(29,53,141,82),(30,62,142,91),(31,55,143,84),(32,64,144,93),(33,157,103,122),(34,150,104,115),(35,159,105,124),(36,152,106,117),(37,145,107,126),(38,154,108,119),(39,147,109,128),(40,156,110,121),(41,149,111,114),(42,158,112,123),(43,151,97,116),(44,160,98,125),(45,153,99,118),(46,146,100,127),(47,155,101,120),(48,148,102,113)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 12I | ··· | 12P | 16A | ··· | 16P | 24A | ··· | 24P | 24Q | ··· | 24X | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C12 | C24 | C24 | M5(2) | C3×M5(2) |
| kernel | C3×C16⋊5C4 | C4×C24 | C2×C48 | C16⋊5C4 | C48 | C4×C12 | C2×C24 | C4×C8 | C2×C16 | C24 | C2×C12 | C16 | C42 | C2×C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 4 | 4 | 16 | 16 | 8 | 16 |
Matrix representation of C3×C16⋊5C4 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 75 | 0 | 0 |
| 0 | 19 | 88 |
| 0 | 78 | 78 |
| 75 | 0 | 0 |
| 0 | 96 | 95 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,35],[75,0,0,0,19,78,0,88,78],[75,0,0,0,96,0,0,95,1] >;
C3×C16⋊5C4 in GAP, Magma, Sage, TeX
C_3\times C_{16}\rtimes_5C_4 % in TeX
G:=Group("C3xC16:5C4"); // GroupNames label
G:=SmallGroup(192,152);
// by ID
G=gap.SmallGroup(192,152);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,1373,176,136,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
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