direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×M5(2), C4.C24, C48⋊7C2, C16⋊3C6, C12.4C8, C24.6C4, C8.2C12, C22.C24, C24.29C22, C8.8(C2×C6), (C2×C8).8C6, (C2×C6).1C8, C6.13(C2×C8), (C2×C4).5C12, C2.3(C2×C24), C12.49(C2×C4), (C2×C12).14C4, (C2×C24).18C2, C4.12(C2×C12), SmallGroup(96,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M5(2)
G = < a,b,c | a3=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C3×M5(2)
►On 48 points
(1 32 40)(2 17 41)(3 18 42)(4 19 43)(5 20 44)(6 21 45)(7 22 46)(8 23 47)(9 24 48)(10 25 33)(11 26 34)(12 27 35)(13 28 36)(14 29 37)(15 30 38)(16 31 39)
(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 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)
G:=sub<Sym(48)| (1,32,40)(2,17,41)(3,18,42)(4,19,43)(5,20,44)(6,21,45)(7,22,46)(8,23,47)(9,24,48)(10,25,33)(11,26,34)(12,27,35)(13,28,36)(14,29,37)(15,30,38)(16,31,39), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47)>;
G:=Group( (1,32,40)(2,17,41)(3,18,42)(4,19,43)(5,20,44)(6,21,45)(7,22,46)(8,23,47)(9,24,48)(10,25,33)(11,26,34)(12,27,35)(13,28,36)(14,29,37)(15,30,38)(16,31,39), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47) );
G=PermutationGroup([[(1,32,40),(2,17,41),(3,18,42),(4,19,43),(5,20,44),(6,21,45),(7,22,46),(8,23,47),(9,24,48),(10,25,33),(11,26,34),(12,27,35),(13,28,36),(14,29,37),(15,30,38),(16,31,39)], [(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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(33,41),(35,43),(37,45),(39,47)]])
C3×M5(2) is a maximal subgroup of
C24.97D4 C48⋊C4 C24.Q8 C8.25D12 Dic6.C8 M5(2)⋊S3 C12.4D8 D24⋊2C4 C16.12D6 C16⋊D6 C16.D6
60 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | M5(2) | C3×M5(2) |
| kernel | C3×M5(2) | C48 | C2×C24 | M5(2) | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C3×M5(2) ►in GL3(𝔽97) generated by
| 35 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 33 | 95 |
| 0 | 36 | 64 |
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 33 | 96 |
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[1,0,0,0,33,36,0,95,64],[96,0,0,0,1,33,0,0,96] >;
C3×M5(2) in GAP, Magma, Sage, TeX
C_3\times M_5(2)
% in TeX
G:=Group("C3xM5(2)"); // GroupNames label
G:=SmallGroup(96,60);
// by ID
G=gap.SmallGroup(96,60);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,72,601,69,88]);
// Polycyclic
G:=Group<a,b,c|a^3=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations
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