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G = C10xC22:A4order 480 = 25·3·5

Direct product of C10 and C22:A4

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

Aliases: C10xC22:A4, C25:4C15, C24:6C30, C22:(C10xA4), C23:3(C5xA4), (C24xC10):2C3, (C22xC10):3A4, (C23xC10):8C6, (C2xC10):2(C2xA4), SmallGroup(480,1209)

Series: Derived Chief Lower central Upper central

C1C24 — C10xC22:A4
C1C22C24C23xC10C5xC22:A4 — C10xC22:A4
C24 — C10xC22:A4
C1C10

Generators and relations for C10xC22:A4
 G = < a,b,c,d,e,f | a10=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 896 in 296 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C23, C23, C10, C10, A4, C15, C24, C24, C2xC10, C2xC10, C2xA4, C30, C25, C22xC10, C22xC10, C22:A4, C5xA4, C23xC10, C23xC10, C2xC22:A4, C10xA4, C24xC10, C5xC22:A4, C10xC22:A4
Quotients: C1, C2, C3, C5, C6, C10, A4, C15, C2xA4, C30, C22:A4, C5xA4, C2xC22:A4, C10xA4, C5xC22:A4, C10xC22:A4

Smallest permutation representation of C10xC22:A4
On 60 points
Generators in S60
(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)
(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 51)(18 52)(19 53)(20 54)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 21)(8 22)(9 23)(10 24)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 30)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 41)
(1 30)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 60)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)
(1 59 39)(2 60 40)(3 51 31)(4 52 32)(5 53 33)(6 54 34)(7 55 35)(8 56 36)(9 57 37)(10 58 38)(11 41 21)(12 42 22)(13 43 23)(14 44 24)(15 45 25)(16 46 26)(17 47 27)(18 48 28)(19 49 29)(20 50 30)

G:=sub<Sym(60)| (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), (11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,41), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,60)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59), (1,59,39)(2,60,40)(3,51,31)(4,52,32)(5,53,33)(6,54,34)(7,55,35)(8,56,36)(9,57,37)(10,58,38)(11,41,21)(12,42,22)(13,43,23)(14,44,24)(15,45,25)(16,46,26)(17,47,27)(18,48,28)(19,49,29)(20,50,30)>;

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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60), (11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,41), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,60)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59), (1,59,39)(2,60,40)(3,51,31)(4,52,32)(5,53,33)(6,54,34)(7,55,35)(8,56,36)(9,57,37)(10,58,38)(11,41,21)(12,42,22)(13,43,23)(14,44,24)(15,45,25)(16,46,26)(17,47,27)(18,48,28)(19,49,29)(20,50,30) );

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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60)], [(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,51),(18,52),(19,53),(20,54),(31,47),(32,48),(33,49),(34,50),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,21),(8,22),(9,23),(10,24),(31,47),(32,48),(33,49),(34,50),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,30),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50),(40,41)], [(1,30),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,60),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59)], [(1,59,39),(2,60,40),(3,51,31),(4,52,32),(5,53,33),(6,54,34),(7,55,35),(8,56,36),(9,57,37),(10,58,38),(11,41,21),(12,42,22),(13,43,23),(14,44,24),(15,45,25),(16,46,26),(17,47,27),(18,48,28),(19,49,29),(20,50,30)]])

80 conjugacy classes

class 1 2A2B···2K3A3B5A5B5C5D6A6B10A10B10C10D10E···10AR15A···15H30A···30H
order122···2335555661010101010···1015···1530···30
size113···316161111161611113···316···1616···16

80 irreducible representations

dim111111113333
type++++
imageC1C2C3C5C6C10C15C30A4C2xA4C5xA4C10xA4
kernelC10xC22:A4C5xC22:A4C24xC10C2xC22:A4C23xC10C22:A4C25C24C22xC10C2xC10C23C22
# reps11242488552020

Matrix representation of C10xC22:A4 in GL6(F31)

2700000
0270000
0027000
000400
000040
000004
,
100000
5300000
25030000
000100
0000300
00030030
,
3000000
0300000
601000
0003000
0000300
000111
,
100000
010000
001000
0003000
0000300
000111
,
100000
010000
001000
0003000
000010
00003030
,
5290000
0261000
060000
000010
000303029
000001

G:=sub<GL(6,GF(31))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,5,25,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,30,0,0,0,0,30,0,0,0,0,0,0,30],[30,0,6,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,1,0,0,0,0,30,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,1,0,0,0,0,30,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,1,30,0,0,0,0,0,30],[5,0,0,0,0,0,29,26,6,0,0,0,0,1,0,0,0,0,0,0,0,0,30,0,0,0,0,1,30,0,0,0,0,0,29,1] >;

C10xC22:A4 in GAP, Magma, Sage, TeX

C_{10}\times C_2^2\rtimes A_4
% in TeX

G:=Group("C10xC2^2:A4");
// GroupNames label

G:=SmallGroup(480,1209);
// by ID

G=gap.SmallGroup(480,1209);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,850,1586,5052,8833]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^10=b^2=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,f*c*f^-1=b,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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