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G = C2xA4xF5order 480 = 25·3·5

Direct product of C2, A4 and F5

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

Aliases: C2xA4xF5, C10:(C4xA4), D5:(C4xA4), (C23xF5):C3, C22:(C6xF5), (D5xA4):3C4, (C10xA4):2C4, (C22xF5):C6, (C22xC10):C12, (C22xD5):C12, (C23xD5).C6, D5.(C22xA4), C23:2(C3xF5), D10.4(C2xA4), (D5xA4).3C22, C5:(C2xC4xA4), (C2xC10):(C2xC12), (C2xD5xA4).3C2, (C5xA4):3(C2xC4), (C22xD5).(C2xC6), SmallGroup(480,1192)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C2xA4xF5
C1C5C2xC10C22xD5D5xA4A4xF5 — C2xA4xF5
C2xC10 — C2xA4xF5
C1C2

Generators and relations for C2xA4xF5
 G = < a,b,c,d,e,f | a2=b2=c2=d3=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=bc=cb, be=eb, bf=fb, dcd-1=b, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 844 in 132 conjugacy classes, 30 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2xC4, C23, C23, D5, D5, C10, C10, C12, A4, C2xC6, C15, C22xC4, C24, F5, F5, D10, D10, C2xC10, C2xC10, C2xC12, C2xA4, C2xA4, C3xD5, C30, C23xC4, C2xF5, C2xF5, C22xD5, C22xD5, C22xC10, C4xA4, C22xA4, C3xF5, C5xA4, C6xD5, C22xF5, C22xF5, C23xD5, C2xC4xA4, D5xA4, C6xF5, C10xA4, C23xF5, A4xF5, C2xD5xA4, C2xA4xF5
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C12, A4, C2xC6, F5, C2xC12, C2xA4, C2xF5, C4xA4, C22xA4, C3xF5, C2xC4xA4, C6xF5, A4xF5, C2xA4xF5

Permutation representations of C2xA4xF5
On 30 points - transitive group 30T107
Generators in S30
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)
(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 8 5 9)(3 10 4 7)(11 16)(12 18 15 19)(13 20 14 17)(21 26)(22 28 25 29)(23 30 24 27)

G:=sub<Sym(30)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20), (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)(2,8,5,9)(3,10,4,7)(11,16)(12,18,15,19)(13,20,14,17)(21,26)(22,28,25,29)(23,30,24,27)>;

G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20), (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)(2,8,5,9)(3,10,4,7)(11,16)(12,18,15,19)(13,20,14,17)(21,26)(22,28,25,29)(23,30,24,27) );

G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,6),(2,7),(3,8),(4,9),(5,10),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,21,11),(2,22,12),(3,23,13),(4,24,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20)], [(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),(2,8,5,9),(3,10,4,7),(11,16),(12,18,15,19),(13,20,14,17),(21,26),(22,28,25,29),(23,30,24,27)]])

G:=TransitiveGroup(30,107);

40 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H 5 6A6B6C6D6E6F10A10B10C12A···12H15A15B30A30B
order122222223344444444566666610101012···1215153030
size113355151544555515151515444202020204121220···2016161616

40 irreducible representations

dim11111111111212333334444
type++++++++++
imageC1C2C2C3C4C4C6C6C12C12A4xF5C2xA4xF5A4C2xA4C2xA4C4xA4C4xA4F5C2xF5C3xF5C6xF5
kernelC2xA4xF5A4xF5C2xD5xA4C23xF5D5xA4C10xA4C22xF5C23xD5C22xD5C22xC10C2C1C2xF5F5D10D5C10C2xA4A4C23C22
# reps121222424411121221122

Matrix representation of C2xA4xF5 in GL7(F61)

60000000
06000000
00600000
0001000
0000100
0000010
0000001
,
60000000
06000000
14010000
0001000
0000100
0000010
0000001
,
60000000
48100000
00600000
0001000
0000100
0000010
0000001
,
135900000
04810000
01400000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
00060606060
0001000
0000100
0000010
,
50000000
05000000
00500000
0001000
0000001
0000100
00060606060

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,0,14,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,48,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,59,48,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,1,0,0,0,0,60,0,0,1,0,0,0,60,0,0,0],[50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,1,0,60] >;

C2xA4xF5 in GAP, Magma, Sage, TeX

C_2\times A_4\times F_5
% in TeX

G:=Group("C2xA4xF5");
// GroupNames label

G:=SmallGroup(480,1192);
// by ID

G=gap.SmallGroup(480,1192);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,648,271,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^5=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

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