direct product, non-abelian, soluble, monomial
Aliases: C10×S4, (C2×A4)⋊C10, A4⋊(C2×C10), C23⋊(C5×S3), (C2×C10)⋊2D6, C22⋊(S3×C10), (C10×A4)⋊3C2, (C5×A4)⋊4C22, (C22×C10)⋊1S3, SmallGroup(240,196)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C10×S4 |
Generators and relations for C10×S4
G = < a,b,c,d,e | a10=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Subgroups: 196 in 66 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, C10, C10, A4, D6, C15, C2×D4, C20, C2×C10, C2×C10, S4, C2×A4, C5×S3, C30, C2×C20, C5×D4, C22×C10, C22×C10, C2×S4, C5×A4, S3×C10, D4×C10, C5×S4, C10×A4, C10×S4
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, C5×S4, C10×S4
►On 30 points - transitive group 30T65
(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)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 28 17)(2 29 18)(3 30 19)(4 21 20)(5 22 11)(6 23 12)(7 24 13)(8 25 14)(9 26 15)(10 27 16)
(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 21)
G:=sub<Sym(30)| (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), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,28,17)(2,29,18)(3,30,19)(4,21,20)(5,22,11)(6,23,12)(7,24,13)(8,25,14)(9,26,15)(10,27,16), (11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21)>;
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), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,28,17)(2,29,18)(3,30,19)(4,21,20)(5,22,11)(6,23,12)(7,24,13)(8,25,14)(9,26,15)(10,27,16), (11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21) );
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)], [(1,6),(2,7),(3,8),(4,9),(5,10),(21,26),(22,27),(23,28),(24,29),(25,30)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,28,17),(2,29,18),(3,30,19),(4,21,20),(5,22,11),(6,23,12),(7,24,13),(8,25,14),(9,26,15),(10,27,16)], [(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,21)]])
G:=TransitiveGroup(30,65);
C10×S4 is a maximal subgroup of
Dic5⋊S4 D10⋊S4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 10M | ··· | 10T | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | 30B | 30C | 30D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | 30 | 30 | 30 |
| size | 1 | 1 | 3 | 3 | 6 | 6 | 8 | 6 | 6 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | C5×S3 | S3×C10 | S4 | C2×S4 | C5×S4 | C10×S4 |
| kernel | C10×S4 | C5×S4 | C10×A4 | C2×S4 | S4 | C2×A4 | C22×C10 | C2×C10 | C23 | C22 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 |
Matrix representation of C10×S4 ►in GL3(𝔽11) generated by
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 10 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 10 |
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 1 |
| 0 | 0 | 8 |
| 2 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 5 | 0 |
| 9 | 0 | 0 |
| 0 | 0 | 10 |
G:=sub<GL(3,GF(11))| [2,0,0,0,2,0,0,0,2],[10,0,0,0,1,0,0,0,10],[10,0,0,0,10,0,0,0,1],[0,2,0,0,0,9,8,0,0],[0,9,0,5,0,0,0,0,10] >;
C10×S4 in GAP, Magma, Sage, TeX
C_{10}\times S_4 % in TeX
G:=Group("C10xS4"); // GroupNames label
G:=SmallGroup(240,196);
// by ID
G=gap.SmallGroup(240,196);
# by ID
G:=PCGroup([6,-2,-2,-5,-3,-2,2,963,3604,202,2165,347]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations