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G = C2xD5xA4order 240 = 24·3·5

Direct product of C2, D5 and A4

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

Aliases: C2xD5xA4, C10:(C2xA4), C5:(C22xA4), (C23xD5):C3, C23:(C3xD5), C22:(C6xD5), (C22xC10):C6, (C22xD5):C6, (C10xA4):2C2, (C5xA4):3C22, (C2xC10):(C2xC6), SmallGroup(240,198)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C2xD5xA4
C1C5C2xC10C5xA4D5xA4 — C2xD5xA4
C2xC10 — C2xD5xA4
C1C2

Generators and relations for C2xD5xA4
 G = < a,b,c,d,e,f | a2=b5=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 448 in 78 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C23, C23, D5, D5, C10, C10, A4, C2xC6, C15, C24, D10, D10, C2xC10, C2xC10, C2xA4, C2xA4, C3xD5, C30, C22xD5, C22xD5, C22xC10, C22xA4, C5xA4, C6xD5, C23xD5, D5xA4, C10xA4, C2xD5xA4
Quotients: C1, C2, C3, C22, C6, D5, A4, C2xC6, D10, C2xA4, C3xD5, C22xA4, C6xD5, D5xA4, C2xD5xA4

Permutation representations of C2xD5xA4
On 30 points - transitive group 30T55
Generators in S30
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 9)(2 10)(3 6)(4 7)(5 8)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)

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

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

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

G:=TransitiveGroup(30,55);

C2xD5xA4 is a maximal subgroup of   D10:S4  A4:D20
C2xD5xA4 is a maximal quotient of   SL2(F3).11D10  Dic10.A4  D20.A4

32 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B5A5B6A6B6C6D6E6F10A10B10C10D10E10F15A15B15C15D30A30B30C30D
order1222222233556666661010101010101515151530303030
size11335515154422442020202022666688888888

32 irreducible representations

dim111111222233366
type++++++++++
imageC1C2C2C3C6C6D5D10C3xD5C6xD5A4C2xA4C2xA4D5xA4C2xD5xA4
kernelC2xD5xA4D5xA4C10xA4C23xD5C22xD5C22xC10C2xA4A4C23C22D10D5C10C2C1
# reps121242224412122

Matrix representation of C2xD5xA4 in GL5(F31)

300000
030000
00100
00010
00001
,
01000
3018000
00100
00010
00001
,
01000
10000
00100
00010
00001
,
10000
01000
00100
0019300
0029030
,
10000
01000
003000
001210
000030
,
250000
025000
0025025
00005
000306

G:=sub<GL(5,GF(31))| [30,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,30,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,19,29,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,30,12,0,0,0,0,1,0,0,0,0,0,30],[25,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,0,30,0,0,25,5,6] >;

C2xD5xA4 in GAP, Magma, Sage, TeX

C_2\times D_5\times A_4
% in TeX

G:=Group("C2xD5xA4");
// GroupNames label

G:=SmallGroup(240,198);
// by ID

G=gap.SmallGroup(240,198);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-5,231,106,6917]);
// Polycyclic

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

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