C2^2xC2^4:C5 - GroupNames

G = C₂²×C₂⁴⋊C₅ order 320 = 2⁶·5

Direct product of C₂² and C₂⁴⋊C₅

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

Aliases: C₂²×C₂⁴⋊C₅, C₂⁶⋊C₅, C₂⁵⋊C₁₀, C₂⁴⋊(C₂×C₁₀), SmallGroup(320,1637)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂²×C₂⁴⋊C₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂×C₂⁴⋊C₅ — C₂²×C₂⁴⋊C₅

Lower central

C₂⁴ — C₂²×C₂⁴⋊C₅

Upper central

C₁ — C₂²

Generators and relations for C₂²×C₂⁴⋊C₅
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=g⁵=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg^-1=cde, de=ed, df=fd, gdg^-1=def, geg^-1=ef=fe, gfg^-1=c >

Subgroups: 2910 in 583 conjugacy classes, 15 normal (6 characteristic)
C₁, C₂, C₂, C₂², C₂², C₅, C₂³, C₁₀, C₂⁴, C₂⁴, C₂×C₁₀, C₂⁵, C₂⁵, C₂⁶, C₂⁴⋊C₅, C₂×C₂⁴⋊C₅, C₂²×C₂⁴⋊C₅
Quotients: C₁, C₂, C₂², C₅, C₁₀, C₂×C₁₀, C₂⁴⋊C₅, C₂×C₂⁴⋊C₅, C₂²×C₂⁴⋊C₅

Permutation representations of C₂²×C₂⁴⋊C₅
►On 20 points - transitive group 20T72

Generators in S₂₀

(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)

(1 11)(2 17)(3 18)(4 14)(6 16)(7 12)(8 13)(9 19)

(2 7)(3 13)(4 14)(5 10)(8 18)(9 19)(12 17)(15 20)

(1 6)(2 7)(3 18)(5 20)(8 13)(10 15)(11 16)(12 17)

(2 12)(3 18)(4 19)(5 15)(7 17)(8 13)(9 14)(10 20)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

G:=sub<Sym(20)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,11)(2,17)(3,18)(4,14)(6,16)(7,12)(8,13)(9,19), (2,7)(3,13)(4,14)(5,10)(8,18)(9,19)(12,17)(15,20), (1,6)(2,7)(3,18)(5,20)(8,13)(10,15)(11,16)(12,17), (2,12)(3,18)(4,19)(5,15)(7,17)(8,13)(9,14)(10,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;

G:=Group( (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,11)(2,17)(3,18)(4,14)(6,16)(7,12)(8,13)(9,19), (2,7)(3,13)(4,14)(5,10)(8,18)(9,19)(12,17)(15,20), (1,6)(2,7)(3,18)(5,20)(8,13)(10,15)(11,16)(12,17), (2,12)(3,18)(4,19)(5,15)(7,17)(8,13)(9,14)(10,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20) );

G=PermutationGroup([[(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)], [(1,11),(2,17),(3,18),(4,14),(6,16),(7,12),(8,13),(9,19)], [(2,7),(3,13),(4,14),(5,10),(8,18),(9,19),(12,17),(15,20)], [(1,6),(2,7),(3,18),(5,20),(8,13),(10,15),(11,16),(12,17)], [(2,12),(3,18),(4,19),(5,15),(7,17),(8,13),(9,14),(10,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)]])

G:=TransitiveGroup(20,72);

►On 20 points - transitive group 20T74

Generators in S₂₀

(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)

(1 6)(4 14)(5 20)(9 19)(10 15)(11 16)

(2 12)(3 8)(4 14)(5 10)(7 17)(9 19)(13 18)(15 20)

(1 6)(2 12)(3 18)(4 19)(5 20)(7 17)(8 13)(9 14)(10 15)(11 16)

(1 16)(2 7)(5 15)(6 11)(10 20)(12 17)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

G:=sub<Sym(20)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,6)(4,14)(5,20)(9,19)(10,15)(11,16), (2,12)(3,8)(4,14)(5,10)(7,17)(9,19)(13,18)(15,20), (1,6)(2,12)(3,18)(4,19)(5,20)(7,17)(8,13)(9,14)(10,15)(11,16), (1,16)(2,7)(5,15)(6,11)(10,20)(12,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;

G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,6)(4,14)(5,20)(9,19)(10,15)(11,16), (2,12)(3,8)(4,14)(5,10)(7,17)(9,19)(13,18)(15,20), (1,6)(2,12)(3,18)(4,19)(5,20)(7,17)(8,13)(9,14)(10,15)(11,16), (1,16)(2,7)(5,15)(6,11)(10,20)(12,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20) );

G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)], [(1,6),(4,14),(5,20),(9,19),(10,15),(11,16)], [(2,12),(3,8),(4,14),(5,10),(7,17),(9,19),(13,18),(15,20)], [(1,6),(2,12),(3,18),(4,19),(5,20),(7,17),(8,13),(9,14),(10,15),(11,16)], [(1,16),(2,7),(5,15),(6,11),(10,20),(12,17)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)]])

G:=TransitiveGroup(20,74);

►On 20 points - transitive group 20T86

Generators in S₂₀

(1 13)(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)(10 19)

(1 17)(2 18)(3 19)(4 20)(5 16)(6 11)(7 12)(8 13)(9 14)(10 15)

(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(9 18)(10 19)

(4 11)(5 12)(6 20)(7 16)

(1 13)(4 11)(6 20)(8 17)

(1 13)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(10 19)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

G:=sub<Sym(20)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19), (1,17)(2,18)(3,19)(4,20)(5,16)(6,11)(7,12)(8,13)(9,14)(10,15), (2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(9,18)(10,19), (4,11)(5,12)(6,20)(7,16), (1,13)(4,11)(6,20)(8,17), (1,13)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(10,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;

G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19), (1,17)(2,18)(3,19)(4,20)(5,16)(6,11)(7,12)(8,13)(9,14)(10,15), (2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(9,18)(10,19), (4,11)(5,12)(6,20)(7,16), (1,13)(4,11)(6,20)(8,17), (1,13)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(10,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20) );

G=PermutationGroup([[(1,13),(2,14),(3,15),(4,11),(5,12),(6,20),(7,16),(8,17),(9,18),(10,19)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,11),(7,12),(8,13),(9,14),(10,15)], [(2,14),(3,15),(4,11),(5,12),(6,20),(7,16),(9,18),(10,19)], [(4,11),(5,12),(6,20),(7,16)], [(1,13),(4,11),(6,20),(8,17)], [(1,13),(3,15),(4,11),(5,12),(6,20),(7,16),(8,17),(10,19)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)]])

G:=TransitiveGroup(20,86);

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2O	5A	5B	5C	5D	10A	···	10L
order	1	2	2	2	2	···	2	5	5	5	5	10	···	10
size	1	1	1	1	5	···	5	16	16	16	16	16	···	16

32 irreducible representations

dim	1	1	1	1	5	5
type	+	+			+	+
image	C₁	C₂	C₅	C₁₀	C₂⁴⋊C₅	C₂×C₂⁴⋊C₅
kernel	C₂²×C₂⁴⋊C₅	C₂×C₂⁴⋊C₅	C₂⁶	C₂⁵	C₂²	C₂
# reps	1	3	4	12	3	9

Matrix representation of C₂²×C₂⁴⋊C₅ ►in GL₆(𝔽₁₁)

10	0	0	0	0	0
0	10	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	10	0
0	0	0	0	0	10

1	0	0	0	0	0
0	10	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	10	0
0	0	0	0	0	10

1	0	0	0	0	0
0	1	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	1	0
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	10	0	0
0	0	0	0	1	0
0	0	0	0	0	10

1	0	0	0	0	0
0	1	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	10	0
0	0	0	0	0	10

1	0	0	0	0	0
0	10	0	0	0	0
0	0	10	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	1	0	0	0	0

G:=sub<GL(6,GF(11))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C₂²×C₂⁴⋊C₅ in GAP, Magma, Sage, TeX

C_2^2\times C_2^4\rtimes C_5

% in TeX

G:=Group("C2^2xC2^4:C5");

// GroupNames label

G:=SmallGroup(320,1637);

// by ID

G=gap.SmallGroup(320,1637);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,2,2,2,1137,1593,2329,3695]);

// Polycyclic

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

// generators/relations

G = C22×C24⋊C5 order 320 = 26·5

Direct product of C22 and C24⋊C5

G = C₂²×C₂⁴⋊C₅ order 320 = 2⁶·5

Direct product of C₂² and C₂⁴⋊C₅