direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C19, C4.C76, C8⋊3C38, C152⋊7C2, C76.4C4, C22.C76, C76.22C22, C2.3(C2×C76), (C2×C76).8C2, (C2×C38).1C4, (C2×C4).2C38, C4.6(C2×C38), C38.12(C2×C4), SmallGroup(304,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C19
G = < a,b,c | a19=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of M4(2)×C19
►On 152 points
Generators in S152
(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)
(1 45 136 59 133 36 89 96)(2 46 137 60 115 37 90 97)(3 47 138 61 116 38 91 98)(4 48 139 62 117 20 92 99)(5 49 140 63 118 21 93 100)(6 50 141 64 119 22 94 101)(7 51 142 65 120 23 95 102)(8 52 143 66 121 24 77 103)(9 53 144 67 122 25 78 104)(10 54 145 68 123 26 79 105)(11 55 146 69 124 27 80 106)(12 56 147 70 125 28 81 107)(13 57 148 71 126 29 82 108)(14 39 149 72 127 30 83 109)(15 40 150 73 128 31 84 110)(16 41 151 74 129 32 85 111)(17 42 152 75 130 33 86 112)(18 43 134 76 131 34 87 113)(19 44 135 58 132 35 88 114)
(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(58 114)(59 96)(60 97)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)
G:=sub<Sym(152)| (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), (1,45,136,59,133,36,89,96)(2,46,137,60,115,37,90,97)(3,47,138,61,116,38,91,98)(4,48,139,62,117,20,92,99)(5,49,140,63,118,21,93,100)(6,50,141,64,119,22,94,101)(7,51,142,65,120,23,95,102)(8,52,143,66,121,24,77,103)(9,53,144,67,122,25,78,104)(10,54,145,68,123,26,79,105)(11,55,146,69,124,27,80,106)(12,56,147,70,125,28,81,107)(13,57,148,71,126,29,82,108)(14,39,149,72,127,30,83,109)(15,40,150,73,128,31,84,110)(16,41,151,74,129,32,85,111)(17,42,152,75,130,33,86,112)(18,43,134,76,131,34,87,113)(19,44,135,58,132,35,88,114), (20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(58,114)(59,96)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)>;
G:=Group( (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), (1,45,136,59,133,36,89,96)(2,46,137,60,115,37,90,97)(3,47,138,61,116,38,91,98)(4,48,139,62,117,20,92,99)(5,49,140,63,118,21,93,100)(6,50,141,64,119,22,94,101)(7,51,142,65,120,23,95,102)(8,52,143,66,121,24,77,103)(9,53,144,67,122,25,78,104)(10,54,145,68,123,26,79,105)(11,55,146,69,124,27,80,106)(12,56,147,70,125,28,81,107)(13,57,148,71,126,29,82,108)(14,39,149,72,127,30,83,109)(15,40,150,73,128,31,84,110)(16,41,151,74,129,32,85,111)(17,42,152,75,130,33,86,112)(18,43,134,76,131,34,87,113)(19,44,135,58,132,35,88,114), (20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(58,114)(59,96)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113) );
G=PermutationGroup([[(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)], [(1,45,136,59,133,36,89,96),(2,46,137,60,115,37,90,97),(3,47,138,61,116,38,91,98),(4,48,139,62,117,20,92,99),(5,49,140,63,118,21,93,100),(6,50,141,64,119,22,94,101),(7,51,142,65,120,23,95,102),(8,52,143,66,121,24,77,103),(9,53,144,67,122,25,78,104),(10,54,145,68,123,26,79,105),(11,55,146,69,124,27,80,106),(12,56,147,70,125,28,81,107),(13,57,148,71,126,29,82,108),(14,39,149,72,127,30,83,109),(15,40,150,73,128,31,84,110),(16,41,151,74,129,32,85,111),(17,42,152,75,130,33,86,112),(18,43,134,76,131,34,87,113),(19,44,135,58,132,35,88,114)], [(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(58,114),(59,96),(60,97),(61,98),(62,99),(63,100),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113)]])
190 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38AJ | 76A | ··· | 76AJ | 76AK | ··· | 76BB | 152A | ··· | 152BT |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
190 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C19 | C38 | C38 | C76 | C76 | M4(2) | M4(2)×C19 |
| kernel | M4(2)×C19 | C152 | C2×C76 | C76 | C2×C38 | M4(2) | C8 | C2×C4 | C4 | C22 | C19 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 18 | 36 | 18 | 36 | 36 | 2 | 36 |
Matrix representation of M4(2)×C19 ►in GL2(𝔽457) generated by
| 256 | 0 |
| 0 | 256 |
| 0 | 1 |
| 348 | 0 |
| 1 | 0 |
| 0 | 456 |
G:=sub<GL(2,GF(457))| [256,0,0,256],[0,348,1,0],[1,0,0,456] >;
M4(2)×C19 in GAP, Magma, Sage, TeX
M_4(2)\times C_{19} % in TeX
G:=Group("M4(2)xC19"); // GroupNames label
G:=SmallGroup(304,23);
// by ID
G=gap.SmallGroup(304,23);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,380,1541,58]);
// Polycyclic
G:=Group<a,b,c|a^19=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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