C3xC2^4:C5 - GroupNames

G = C₃×C₂⁴⋊C₅ order 240 = 2⁴·3·5

Direct product of C₃ and C₂⁴⋊C₅

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

Aliases: C₃×C₂⁴⋊C₅, C₂⁴⋊₂C₁₅, (C₂³×C₆)⋊C₅, SmallGroup(240,199)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₃×C₂⁴⋊C₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₃×C₂⁴⋊C₅

Lower central

C₂⁴ — C₃×C₂⁴⋊C₅

Upper central

C₁ — C₃

Generators and relations for C₃×C₂⁴⋊C₅
G = < a,b,c,d,e,f | a³=b²=c²=d²=e²=f⁵=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bcd, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, fef^-1=b >

C₁

₅C₂

C₃

₁₆C₅

₅C₂²

₅C₆

₁₆C₁₅

₅C₂³

₅C₂×C₆

C₂⁴

₅C₂²×C₆

C₂³×C₆

C₂⁴⋊C₅

C₃×C₂⁴⋊C₅

Character table of C₃×C₂⁴⋊C₅

class	1	2A	2B	2C	3A	3B	5A	5B	5C	5D	6A	6B	6C	6D	6E	6F	15A	15B	15C	15D	15E	15F	15G	15H
size	1	5	5	5	1	1	16	16	16	16	5	5	5	5	5	5	16	16	16	16	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃²	ζ₃	linear of order 3
ρ₃	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃	ζ₃²	linear of order 3
ρ₄	1	1	1	1	1	1	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	1	1	1	1	1	1	ζ₅²	ζ₅	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	ζ₅⁴	ζ₅³	linear of order 5
ρ₅	1	1	1	1	1	1	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	1	1	1	1	1	1	ζ₅³	ζ₅⁴	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	ζ₅	ζ₅²	linear of order 5
ρ₆	1	1	1	1	1	1	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	1	1	1	1	1	1	ζ₅⁴	ζ₅²	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	ζ₅³	ζ₅	linear of order 5
ρ₇	1	1	1	1	1	1	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	1	1	1	1	1	1	ζ₅	ζ₅³	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	ζ₅²	ζ₅⁴	linear of order 5
ρ₈	1	1	1	1	ζ₃²	ζ₃	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃ζ₅³	ζ₃²ζ₅⁴	ζ₃²ζ₅	ζ₃²ζ₅³	ζ₃ζ₅²	ζ₃ζ₅⁴	ζ₃ζ₅	ζ₃²ζ₅²	linear of order 15
ρ₉	1	1	1	1	ζ₃²	ζ₃	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃ζ₅⁴	ζ₃²ζ₅²	ζ₃²ζ₅³	ζ₃²ζ₅⁴	ζ₃ζ₅	ζ₃ζ₅²	ζ₃ζ₅³	ζ₃²ζ₅	linear of order 15
ρ₁₀	1	1	1	1	ζ₃²	ζ₃	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃ζ₅²	ζ₃²ζ₅	ζ₃²ζ₅⁴	ζ₃²ζ₅²	ζ₃ζ₅³	ζ₃ζ₅	ζ₃ζ₅⁴	ζ₃²ζ₅³	linear of order 15
ρ₁₁	1	1	1	1	ζ₃	ζ₃²	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²ζ₅³	ζ₃ζ₅⁴	ζ₃ζ₅	ζ₃ζ₅³	ζ₃²ζ₅²	ζ₃²ζ₅⁴	ζ₃²ζ₅	ζ₃ζ₅²	linear of order 15
ρ₁₂	1	1	1	1	ζ₃	ζ₃²	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²ζ₅	ζ₃ζ₅³	ζ₃ζ₅²	ζ₃ζ₅	ζ₃²ζ₅⁴	ζ₃²ζ₅³	ζ₃²ζ₅²	ζ₃ζ₅⁴	linear of order 15
ρ₁₃	1	1	1	1	ζ₃²	ζ₃	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃ζ₅	ζ₃²ζ₅³	ζ₃²ζ₅²	ζ₃²ζ₅	ζ₃ζ₅⁴	ζ₃ζ₅³	ζ₃ζ₅²	ζ₃²ζ₅⁴	linear of order 15
ρ₁₄	1	1	1	1	ζ₃	ζ₃²	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²ζ₅⁴	ζ₃ζ₅²	ζ₃ζ₅³	ζ₃ζ₅⁴	ζ₃²ζ₅	ζ₃²ζ₅²	ζ₃²ζ₅³	ζ₃ζ₅	linear of order 15
ρ₁₅	1	1	1	1	ζ₃	ζ₃²	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²ζ₅²	ζ₃ζ₅	ζ₃ζ₅⁴	ζ₃ζ₅²	ζ₃²ζ₅³	ζ₃²ζ₅	ζ₃²ζ₅⁴	ζ₃ζ₅³	linear of order 15
ρ₁₆	5	1	1	-3	5	5	0	0	0	0	-3	-3	1	1	1	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅
ρ₁₇	5	1	-3	1	5	5	0	0	0	0	1	1	1	1	-3	-3	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅
ρ₁₈	5	-3	1	1	5	5	0	0	0	0	1	1	-3	-3	1	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅
ρ₁₉	5	1	1	-3	^-5-5√-3/₂	^-5+5√-3/₂	0	0	0	0	^3-3√-3/₂	^3+3√-3/₂	ζ₃²	ζ₃	ζ₃	ζ₃²	0	0	0	0	0	0	0	0	complex faithful
ρ₂₀	5	1	-3	1	^-5+5√-3/₂	^-5-5√-3/₂	0	0	0	0	ζ₃²	ζ₃	ζ₃	ζ₃²	^3+3√-3/₂	^3-3√-3/₂	0	0	0	0	0	0	0	0	complex faithful
ρ₂₁	5	-3	1	1	^-5+5√-3/₂	^-5-5√-3/₂	0	0	0	0	ζ₃²	ζ₃	^3-3√-3/₂	^3+3√-3/₂	ζ₃²	ζ₃	0	0	0	0	0	0	0	0	complex faithful
ρ₂₂	5	1	1	-3	^-5+5√-3/₂	^-5-5√-3/₂	0	0	0	0	^3+3√-3/₂	^3-3√-3/₂	ζ₃	ζ₃²	ζ₃²	ζ₃	0	0	0	0	0	0	0	0	complex faithful
ρ₂₃	5	1	-3	1	^-5-5√-3/₂	^-5+5√-3/₂	0	0	0	0	ζ₃	ζ₃²	ζ₃²	ζ₃	^3-3√-3/₂	^3+3√-3/₂	0	0	0	0	0	0	0	0	complex faithful
ρ₂₄	5	-3	1	1	^-5-5√-3/₂	^-5+5√-3/₂	0	0	0	0	ζ₃	ζ₃²	^3+3√-3/₂	^3-3√-3/₂	ζ₃	ζ₃²	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of C₃×C₂⁴⋊C₅
►On 30 points - transitive group 30T52

Generators in S₃₀

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

(1 11)(3 13)(8 27)(10 29)(17 21)(19 23)

(1 11)(2 12)(4 14)(5 15)(6 30)(7 26)(8 27)(9 28)(16 25)(17 21)(18 22)(20 24)

(3 13)(4 14)(6 30)(10 29)(19 23)(20 24)

(2 12)(4 14)(6 30)(9 28)(18 22)(20 24)

(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)

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

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

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

G:=TransitiveGroup(30,52);

C₃×C₂⁴⋊C₅ is a maximal subgroup of C₂⁴⋊D₁₅

Matrix representation of C₃×C₂⁴⋊C₅ ►in GL₅(𝔽₃₁)

5	0	0	0	0
0	5	0	0	0
0	0	5	0	0
0	0	0	5	0
0	0	0	0	5

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
4	20	22	30	0
26	10	8	0	30

1	0	0	0	0
0	30	0	0	0
0	0	1	0	0
0	11	0	1	0
26	0	8	0	30

30	0	0	0	0
0	30	0	0	0
0	0	1	0	0
0	0	22	30	0
0	0	8	0	30

1	0	0	0	0
0	1	0	0	0
0	0	30	0	0
4	20	0	30	0
0	0	23	0	1

0	1	0	0	0
0	0	1	0	0
4	20	22	29	0
0	0	0	9	1
0	0	0	23	0

G:=sub<GL(5,GF(31))| [5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,4,26,0,1,0,20,10,0,0,1,22,8,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,26,0,30,0,11,0,0,0,1,0,8,0,0,0,1,0,0,0,0,0,30],[30,0,0,0,0,0,30,0,0,0,0,0,1,22,8,0,0,0,30,0,0,0,0,0,30],[1,0,0,4,0,0,1,0,20,0,0,0,30,0,23,0,0,0,30,0,0,0,0,0,1],[0,0,4,0,0,1,0,20,0,0,0,1,22,0,0,0,0,29,9,23,0,0,0,1,0] >;

C₃×C₂⁴⋊C₅ in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes C_5

% in TeX

G:=Group("C3xC2^4:C5");

// GroupNames label

G:=SmallGroup(240,199);

// by ID

G=gap.SmallGroup(240,199);

# by ID

G:=PCGroup([6,-3,-5,-2,2,2,2,728,1089,1660,2711]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₃×C₂⁴⋊C₅ in TeX
Character table of C₃×C₂⁴⋊C₅ in TeX

G = C3×C24⋊C5 order 240 = 24·3·5

Direct product of C3 and C24⋊C5

G = C₃×C₂⁴⋊C₅ order 240 = 2⁴·3·5

Direct product of C₃ and C₂⁴⋊C₅