metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C48⋊2C4, C16⋊2Dic3, C12.18C42, M5(2).3S3, C12.5M4(2), C3⋊C16⋊4C4, C8.33(C4×S3), C3⋊2(C16⋊C4), C24.82(C2×C4), (C2×C8).151D6, C6.5(C8⋊C4), C4.9(C8⋊S3), (C4×Dic3).3C4, C24⋊C4.10C2, C4.23(C4×Dic3), C8.20(C2×Dic3), C2.4(C24⋊C4), (C2×C6).4M4(2), C12.C8.8C2, (C3×M5(2)).2C2, (C2×C24).262C22, C22.6(C8⋊S3), (C2×C3⋊C8).1C4, (C2×C12).51(C2×C4), (C2×C4).136(C4×S3), SmallGroup(192,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C48⋊C4
G = < a,b | a48=b4=1, bab-1=a29 >
Smallest permutation representation of C48⋊C4
►On 48 points
Generators in S48
(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)
(2 6 26 30)(3 11)(4 16 28 40)(5 21)(7 31)(8 36 32 12)(9 41)(10 46 34 22)(14 18 38 42)(15 23)(17 33)(19 43)(20 48 44 24)(27 35)(29 45)(39 47)
G:=sub<Sym(48)| (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), (2,6,26,30)(3,11)(4,16,28,40)(5,21)(7,31)(8,36,32,12)(9,41)(10,46,34,22)(14,18,38,42)(15,23)(17,33)(19,43)(20,48,44,24)(27,35)(29,45)(39,47)>;
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), (2,6,26,30)(3,11)(4,16,28,40)(5,21)(7,31)(8,36,32,12)(9,41)(10,46,34,22)(14,18,38,42)(15,23)(17,33)(19,43)(20,48,44,24)(27,35)(29,45)(39,47) );
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)], [(2,6,26,30),(3,11),(4,16,28,40),(5,21),(7,31),(8,36,32,12),(9,41),(10,46,34,22),(14,18,38,42),(15,23),(17,33),(19,43),(20,48,44,24),(27,35),(29,45),(39,47)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 12 | 12 | 2 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | Dic3 | D6 | M4(2) | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | C8⋊S3 | C16⋊C4 | C48⋊C4 |
| kernel | C48⋊C4 | C12.C8 | C24⋊C4 | C3×M5(2) | C3⋊C16 | C48 | C2×C3⋊C8 | C4×Dic3 | M5(2) | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C48⋊C4 ►in GL4(𝔽5) generated by
| 0 | 1 | 1 | 0 |
| 2 | 0 | 0 | 0 |
| 4 | 0 | 0 | 3 |
| 0 | 4 | 2 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 3 | 4 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [0,2,4,0,1,0,0,4,1,0,0,2,0,0,3,0],[4,0,0,0,0,3,0,0,0,4,2,0,0,0,0,1] >;
C48⋊C4 in GAP, Magma, Sage, TeX
C_{48}\rtimes C_4 % in TeX
G:=Group("C48:C4"); // GroupNames label
G:=SmallGroup(192,71);
// by ID
G=gap.SmallGroup(192,71);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,80,102,6278]);
// Polycyclic
G:=Group<a,b|a^48=b^4=1,b*a*b^-1=a^29>;
// generators/relations
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