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## G = C2×D5×A4order 240 = 24·3·5

### Direct product of C2, D5 and A4

Aliases: C2×D5×A4, C10⋊(C2×A4), C5⋊(C22×A4), (C23×D5)⋊C3, C23⋊(C3×D5), C22⋊(C6×D5), (C22×C10)⋊C6, (C22×D5)⋊C6, (C10×A4)⋊2C2, (C5×A4)⋊3C22, (C2×C10)⋊(C2×C6), SmallGroup(240,198)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×D5×A4
 Chief series C1 — C5 — C2×C10 — C5×A4 — D5×A4 — C2×D5×A4
 Lower central C2×C10 — C2×D5×A4
 Upper central C1 — C2

Generators and relations for C2×D5×A4
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, C2×C6, C15, C24, D10, D10, C2×C10, C2×C10, C2×A4, C2×A4, C3×D5, C30, C22×D5, C22×D5, C22×C10, C22×A4, C5×A4, C6×D5, C23×D5, D5×A4, C10×A4, C2×D5×A4
Quotients: C1, C2, C3, C22, C6, D5, A4, C2×C6, D10, C2×A4, C3×D5, C22×A4, C6×D5, D5×A4, C2×D5×A4

Permutation representations of C2×D5×A4
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);

C2×D5×A4 is a maximal subgroup of   D10⋊S4  A4⋊D20
C2×D5×A4 is a maximal quotient of   SL2(𝔽3).11D10  Dic10.A4  D20.A4

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 5A 5B 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 10E 10F 15A 15B 15C 15D 30A 30B 30C 30D order 1 2 2 2 2 2 2 2 3 3 5 5 6 6 6 6 6 6 10 10 10 10 10 10 15 15 15 15 30 30 30 30 size 1 1 3 3 5 5 15 15 4 4 2 2 4 4 20 20 20 20 2 2 6 6 6 6 8 8 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 A4 C2×A4 C2×A4 D5×A4 C2×D5×A4 kernel C2×D5×A4 D5×A4 C10×A4 C23×D5 C22×D5 C22×C10 C2×A4 A4 C23 C22 D10 D5 C10 C2 C1 # reps 1 2 1 2 4 2 2 2 4 4 1 2 1 2 2

Matrix representation of C2×D5×A4 in GL5(𝔽31)

 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 1 0 0 0 30 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 0 0 0 0 19 30 0 0 0 29 0 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 30 0 0 0 0 12 1 0 0 0 0 0 30
,
 25 0 0 0 0 0 25 0 0 0 0 0 25 0 25 0 0 0 0 5 0 0 0 30 6

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] >;

C2×D5×A4 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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