direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C4×C19⋊C6, C76⋊2C6, D19⋊C12, D38.C6, Dic19⋊2C6, (C4×D19)⋊C3, C19⋊C12⋊2C2, C19⋊1(C2×C12), C38.2(C2×C6), C19⋊C3⋊1(C2×C4), (C2×C19⋊C6).C2, (C4×C19⋊C3)⋊2C2, C2.1(C2×C19⋊C6), (C2×C19⋊C3).2C22, SmallGroup(456,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C4×C19⋊C6 |
Generators and relations for C4×C19⋊C6
G = < a,b,c | a4=b19=c6=1, ab=ba, ac=ca, cbc-1=b12 >
Smallest permutation representation of C4×C19⋊C6
►On 76 points
Generators in S76
(1 58 20 39)(2 59 21 40)(3 60 22 41)(4 61 23 42)(5 62 24 43)(6 63 25 44)(7 64 26 45)(8 65 27 46)(9 66 28 47)(10 67 29 48)(11 68 30 49)(12 69 31 50)(13 70 32 51)(14 71 33 52)(15 72 34 53)(16 73 35 54)(17 74 36 55)(18 75 37 56)(19 76 38 57)
(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)
(2 9 8 19 12 13)(3 17 15 18 4 6)(5 14 10 16 7 11)(21 28 27 38 31 32)(22 36 34 37 23 25)(24 33 29 35 26 30)(40 47 46 57 50 51)(41 55 53 56 42 44)(43 52 48 54 45 49)(59 66 65 76 69 70)(60 74 72 75 61 63)(62 71 67 73 64 68)
G:=sub<Sym(76)| (1,58,20,39)(2,59,21,40)(3,60,22,41)(4,61,23,42)(5,62,24,43)(6,63,25,44)(7,64,26,45)(8,65,27,46)(9,66,28,47)(10,67,29,48)(11,68,30,49)(12,69,31,50)(13,70,32,51)(14,71,33,52)(15,72,34,53)(16,73,35,54)(17,74,36,55)(18,75,37,56)(19,76,38,57), (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), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30)(40,47,46,57,50,51)(41,55,53,56,42,44)(43,52,48,54,45,49)(59,66,65,76,69,70)(60,74,72,75,61,63)(62,71,67,73,64,68)>;
G:=Group( (1,58,20,39)(2,59,21,40)(3,60,22,41)(4,61,23,42)(5,62,24,43)(6,63,25,44)(7,64,26,45)(8,65,27,46)(9,66,28,47)(10,67,29,48)(11,68,30,49)(12,69,31,50)(13,70,32,51)(14,71,33,52)(15,72,34,53)(16,73,35,54)(17,74,36,55)(18,75,37,56)(19,76,38,57), (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), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30)(40,47,46,57,50,51)(41,55,53,56,42,44)(43,52,48,54,45,49)(59,66,65,76,69,70)(60,74,72,75,61,63)(62,71,67,73,64,68) );
G=PermutationGroup([[(1,58,20,39),(2,59,21,40),(3,60,22,41),(4,61,23,42),(5,62,24,43),(6,63,25,44),(7,64,26,45),(8,65,27,46),(9,66,28,47),(10,67,29,48),(11,68,30,49),(12,69,31,50),(13,70,32,51),(14,71,33,52),(15,72,34,53),(16,73,35,54),(17,74,36,55),(18,75,37,56),(19,76,38,57)], [(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)], [(2,9,8,19,12,13),(3,17,15,18,4,6),(5,14,10,16,7,11),(21,28,27,38,31,32),(22,36,34,37,23,25),(24,33,29,35,26,30),(40,47,46,57,50,51),(41,55,53,56,42,44),(43,52,48,54,45,49),(59,66,65,76,69,70),(60,74,72,75,61,63),(62,71,67,73,64,68)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 19A | 19B | 19C | 38A | 38B | 38C | 76A | ··· | 76F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 19 | 19 | 19 | 38 | 38 | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 19 | 19 | 19 | 19 | 1 | 1 | 19 | 19 | 19 | ··· | 19 | 19 | ··· | 19 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | C19⋊C6 | C2×C19⋊C6 | C4×C19⋊C6 |
| kernel | C4×C19⋊C6 | C19⋊C12 | C4×C19⋊C3 | C2×C19⋊C6 | C4×D19 | C19⋊C6 | Dic19 | C76 | D38 | D19 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 3 | 3 | 6 |
Matrix representation of C4×C19⋊C6 ►in GL6(𝔽229)
| 122 | 0 | 0 | 0 | 0 | 0 |
| 0 | 122 | 0 | 0 | 0 | 0 |
| 0 | 0 | 122 | 0 | 0 | 0 |
| 0 | 0 | 0 | 122 | 0 | 0 |
| 0 | 0 | 0 | 0 | 122 | 0 |
| 0 | 0 | 0 | 0 | 0 | 122 |
| 91 | 9 | 220 | 138 | 109 | 228 |
| 92 | 9 | 220 | 138 | 109 | 228 |
| 91 | 10 | 220 | 138 | 109 | 228 |
| 91 | 9 | 221 | 138 | 109 | 228 |
| 91 | 9 | 220 | 139 | 109 | 228 |
| 91 | 9 | 220 | 138 | 110 | 228 |
| 220 | 127 | 62 | 148 | 221 | 120 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 148 | 173 | 156 | 192 | 28 | 109 |
| 29 | 81 | 127 | 201 | 139 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 139 | 201 | 127 | 81 | 29 | 110 |
G:=sub<GL(6,GF(229))| [122,0,0,0,0,0,0,122,0,0,0,0,0,0,122,0,0,0,0,0,0,122,0,0,0,0,0,0,122,0,0,0,0,0,0,122],[91,92,91,91,91,91,9,9,10,9,9,9,220,220,220,221,220,220,138,138,138,138,139,138,109,109,109,109,109,110,228,228,228,228,228,228],[220,0,148,29,0,139,127,0,173,81,0,201,62,0,156,127,1,127,148,0,192,201,0,81,221,1,28,139,0,29,120,0,109,1,0,110] >;
C4×C19⋊C6 in GAP, Magma, Sage, TeX
C_4\times C_{19}\rtimes C_6 % in TeX
G:=Group("C4xC19:C6"); // GroupNames label
G:=SmallGroup(456,8);
// by ID
G=gap.SmallGroup(456,8);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,66,10804,1064]);
// Polycyclic
G:=Group<a,b,c|a^4=b^19=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^12>;
// generators/relations
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