direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C4×C19⋊C3, C76⋊C3, C19⋊2C12, C38.2C6, C2.(C2×C19⋊C3), (C2×C19⋊C3).2C2, SmallGroup(228,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C4×C19⋊C3 |
Generators and relations for C4×C19⋊C3
G = < a,b,c | a4=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >
Smallest permutation representation of C4×C19⋊C3
►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 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)(59 65 69)(60 72 61)(62 67 64)(63 74 75)(66 76 70)(68 71 73)
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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)>;
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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73) );
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)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54),(59,65,69),(60,72,61),(62,67,64),(63,74,75),(66,76,70),(68,71,73)]])
C4×C19⋊C3 is a maximal subgroup of
C19⋊C24 Dic38⋊C3 D76⋊C3
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 19A | ··· | 19F | 38A | ··· | 38F | 76A | ··· | 76L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 19 | 19 | 1 | 1 | 19 | 19 | 19 | 19 | 19 | 19 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C19⋊C3 | C2×C19⋊C3 | C4×C19⋊C3 |
| kernel | C4×C19⋊C3 | C2×C19⋊C3 | C76 | C19⋊C3 | C38 | C19 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 |
Matrix representation of C4×C19⋊C3 ►in GL3(𝔽229) generated by
| 122 | 0 | 0 |
| 0 | 122 | 0 |
| 0 | 0 | 122 |
| 96 | 224 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 102 | 223 | 184 |
| 1 | 133 | 5 |
G:=sub<GL(3,GF(229))| [122,0,0,0,122,0,0,0,122],[96,1,0,224,0,1,1,0,0],[1,102,1,0,223,133,0,184,5] >;
C4×C19⋊C3 in GAP, Magma, Sage, TeX
C_4\times C_{19}\rtimes C_3 % in TeX
G:=Group("C4xC19:C3"); // GroupNames label
G:=SmallGroup(228,2);
// by ID
G=gap.SmallGroup(228,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-19,24,679]);
// Polycyclic
G:=Group<a,b,c|a^4=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations
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