Group #68015

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	H(3,𝔽₄)	C₂×C₃
Frattini	C₄²	C₂×A₄
Fitting	#1655	C₃
socle	C₂²	#120⋊C₃
LCS	G ▷ H(3,𝔽₄)
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, cosocle, hypocenter

H(3,𝔽₄)

C₂×C₃

Frattini

C₄²

C₂×A₄

C₃

C₂²

G ▷ H(3,𝔽₄)

C₁

Presentation

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,6,5,9,15)(3,19,22,12,7,24)(4,20,21)(8,17,14)(10,11,23,13,18,16);(1,13)(2,9)(3,5)(4,10)(6,22)(7,11)(12,14)(15,16)(17,24)(21,23)

Polycyclic presentation:

2,3,2^6|e^2=g,f^2=gh,b^a=bh,c^a=ceh,d^a=dfgh,e^a=eg,f^a=fgh,c^b=dg,d^b=cdh,e^b=fh,f^b=efg,g^b=gh,h^b=g,e^c=eg,f^c=fh,e^d=eh,f^d=fgh

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	3	12	16	24	24	24	24	64	64	64	64
order	1	2	4	2	2	4	8	8	3	3	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	-1	1	1	-1	-1	1	1	-1	-1
χ₅	1	1	1	-1	1	1	-1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	1	1	1	-1	1	1	-1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	3	3	3	3	-1	-1	-1	-1	0	0	0	0
χ₈	3	3	3	-3	-1	-1	1	1	0	0	0	0
χ₉	6	6	-2	0	2	-2	0	0	0	0	0	0
χ₁₀	6	6	-2	0	-2	2	0	0	0	0	0	0
χ₁₁	12	-4	0	0	0	0	2	-2	0	0	0	0
χ₁₂	12	-4	0	0	0	0	-2	2	0	0	0	0

Here i is the imaginary unit (i² = −1).

order	384
isoclinism	isoclinic class of #1655⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂×C₄

Fitting, Sylow	#1655 ⋊ C₃
derived	H(3,𝔽₄) ⋊ C₂×C₃

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 24

maximal subgroup of	C₂×#1655⋊C₃, #19073⋊C₃, #19094⋊C₃, …
maximal normal subgroup of	C₂×#1655⋊C₃, #19073⋊C₃, #19094⋊C₃, …
maximal quotient of	#7734⋊C₃, #7749⋊C₃, #7822⋊C₃, …

#1655⋊C₃ #68015

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

#1655⋊C3 #68015

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

#1655⋊C₃ #68015