direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C2×C10, C23⋊C30, C24⋊3C15, C22⋊(C2×C30), (C23×C10)⋊1C3, (C22×C10)⋊2C6, (C2×C10)⋊3(C2×C6), SmallGroup(240,203)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C2×C10 |
Generators and relations for A4×C2×C10
G = < a,b,c,d,e | a2=b10=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Subgroups: 184 in 78 conjugacy classes, 30 normal (12 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C23, C23, C10, C10, A4, C2×C6, C15, C24, C2×C10, C2×C10, C2×A4, C30, C22×C10, C22×C10, C22×A4, C5×A4, C2×C30, C23×C10, C10×A4, A4×C2×C10
Quotients: C1, C2, C3, C22, C5, C6, C10, A4, C2×C6, C15, C2×C10, C2×A4, C30, C22×A4, C5×A4, C2×C30, C10×A4, A4×C2×C10
►On 60 points
Generators in S60
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 21)(9 22)(10 23)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 51)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 41)(38 42)(39 43)(40 44)
(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)
(1 29)(2 30)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 16)(12 17)(13 18)(14 19)(15 20)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 41)(38 42)(39 43)(40 44)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 21)(9 22)(10 23)(11 57)(12 58)(13 59)(14 60)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(1 60 40)(2 51 31)(3 52 32)(4 53 33)(5 54 34)(6 55 35)(7 56 36)(8 57 37)(9 58 38)(10 59 39)(11 46 26)(12 47 27)(13 48 28)(14 49 29)(15 50 30)(16 41 21)(17 42 22)(18 43 23)(19 44 24)(20 45 25)
G:=sub<Sym(60)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,21)(9,22)(10,23)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,51)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,41)(38,42)(39,43)(40,44), (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), (1,29)(2,30)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,16)(12,17)(13,18)(14,19)(15,20)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,41)(38,42)(39,43)(40,44)(51,56)(52,57)(53,58)(54,59)(55,60), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,21)(9,22)(10,23)(11,57)(12,58)(13,59)(14,60)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (1,60,40)(2,51,31)(3,52,32)(4,53,33)(5,54,34)(6,55,35)(7,56,36)(8,57,37)(9,58,38)(10,59,39)(11,46,26)(12,47,27)(13,48,28)(14,49,29)(15,50,30)(16,41,21)(17,42,22)(18,43,23)(19,44,24)(20,45,25)>;
G:=Group( (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,21)(9,22)(10,23)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,51)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,41)(38,42)(39,43)(40,44), (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), (1,29)(2,30)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,16)(12,17)(13,18)(14,19)(15,20)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,41)(38,42)(39,43)(40,44)(51,56)(52,57)(53,58)(54,59)(55,60), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,21)(9,22)(10,23)(11,57)(12,58)(13,59)(14,60)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (1,60,40)(2,51,31)(3,52,32)(4,53,33)(5,54,34)(6,55,35)(7,56,36)(8,57,37)(9,58,38)(10,59,39)(11,46,26)(12,47,27)(13,48,28)(14,49,29)(15,50,30)(16,41,21)(17,42,22)(18,43,23)(19,44,24)(20,45,25) );
G=PermutationGroup([[(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,21),(9,22),(10,23),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,51),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,41),(38,42),(39,43),(40,44)], [(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)], [(1,29),(2,30),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,16),(12,17),(13,18),(14,19),(15,20),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,41),(38,42),(39,43),(40,44),(51,56),(52,57),(53,58),(54,59),(55,60)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,21),(9,22),(10,23),(11,57),(12,58),(13,59),(14,60),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50)], [(1,60,40),(2,51,31),(3,52,32),(4,53,33),(5,54,34),(6,55,35),(7,56,36),(8,57,37),(9,58,38),(10,59,39),(11,46,26),(12,47,27),(13,48,28),(14,49,29),(15,50,30),(16,41,21),(17,42,22),(18,43,23),(19,44,24),(20,45,25)]])
A4×C2×C10 is a maximal subgroup of
C24⋊2D15
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 10M | ··· | 10AB | 15A | ··· | 15H | 30A | ··· | 30X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | A4 | C2×A4 | C5×A4 | C10×A4 |
| kernel | A4×C2×C10 | C10×A4 | C23×C10 | C22×A4 | C22×C10 | C2×A4 | C24 | C23 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 | 1 | 3 | 4 | 12 |
Matrix representation of A4×C2×C10 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 29 | 0 | 0 |
| 0 | 0 | 29 | 0 |
| 0 | 0 | 0 | 29 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,29,0,0,0,0,29,0,0,0,0,29],[1,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,30,0,0,0,0,30,0,0,0,0,1],[5,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
A4×C2×C10 in GAP, Magma, Sage, TeX
A_4\times C_2\times C_{10} % in TeX
G:=Group("A4xC2xC10"); // GroupNames label
G:=SmallGroup(240,203);
// by ID
G=gap.SmallGroup(240,203);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,2,916,1637]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations