C2xD5xA4 - GroupNames

G = C₂×D₅×A₄ order 240 = 2⁴·3·5

Direct product of C₂, D₅ and A₄

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

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

G = C2×D5×A4 order 240 = 24·3·5

Direct product of C2, D5 and A4

G = C₂×D₅×A₄ order 240 = 2⁴·3·5

Direct product of C₂, D₅ and A₄