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G = A4xC2xC10order 240 = 24·3·5

Direct product of C2xC10 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xC2xC10, C23:C30, C24:3C15, C22:(C2xC30), (C23xC10):1C3, (C22xC10):2C6, (C2xC10):3(C2xC6), SmallGroup(240,203)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC2xC10
C1C22C2xC10C5xA4C10xA4 — A4xC2xC10
C22 — A4xC2xC10
C1C2xC10

Generators and relations for A4xC2xC10
 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, C2xC6, C15, C24, C2xC10, C2xC10, C2xA4, C30, C22xC10, C22xC10, C22xA4, C5xA4, C2xC30, C23xC10, C10xA4, A4xC2xC10
Quotients: C1, C2, C3, C22, C5, C6, C10, A4, C2xC6, C15, C2xC10, C2xA4, C30, C22xA4, C5xA4, C2xC30, C10xA4, A4xC2xC10

Smallest permutation representation of A4xC2xC10
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)]])

A4xC2xC10 is a maximal subgroup of   C24:2D15

80 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B5A5B5C5D6A···6F10A···10L10M···10AB15A···15H30A···30X
order122222223355556···610···1010···1015···1530···30
size111133334411114···41···13···34···44···4

80 irreducible representations

dim111111113333
type++++
imageC1C2C3C5C6C10C15C30A4C2xA4C5xA4C10xA4
kernelA4xC2xC10C10xA4C23xC10C22xA4C22xC10C2xA4C24C23C2xC10C10C22C2
# reps132461282413412

Matrix representation of A4xC2xC10 in GL4(F31) generated by

30000
03000
00300
00030
,
1000
02900
00290
00029
,
1000
0100
00300
00030
,
1000
03000
00300
0001
,
5000
0010
0001
0100
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] >;

A4xC2xC10 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

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