Group #68016

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#564	C₂×C₃
Frattini	C₄²	C₂×A₄
Fitting	#1658	C₃
socle	C₂²	#120⋊C₃
LCS	G ▷ #564
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, cosocle, hypocenter

#564

C₂×C₃

Frattini

C₄²

C₂×A₄

C₃

C₂²

G ▷ #564

C₁

Presentation

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,23,6,7,8)(3,34,26,15,29,46)(4,17,28,19,9,40)(5,11,27)(10,44,14,31,25,35)(12,13,45,16,36,37)(18,32,48)(20,21,41)(22,30,47)(24,43,39,33,42,38);(1,14)(2,29,7,13)(3,19)(4,16)(5,35)(6,39)(8,28,23,40)(9,24,17,10)(11,34,21,36)(12,18)(15,22)(20,38)(25,26,42,37)(27,48,41,47)(30,33,32,31)(43,46,44,45)

Polycyclic presentation:

2,3,2^6|c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bgh,c^a=ceg,d^a=dfh,e^a=eg,f^a=fgh,c^b=cdh,d^b=ch,e^b=efgh,f^b=eh,g^b=h,h^b=gh,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	4	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	$2 i$	$- 2 i$	0	0	0	0
χ₁₂	12	-4	0	0	0	0	$- 2 i$	$2 i$	0	0	0	0

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

order	384
isoclinism	isoclinic class of #1658⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Fitting, Sylow	#1658 ⋊ C₃
derived	#564 ⋊ C₂×C₃

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 48

maximal subgroup of	C₂×#1658⋊C₃, #19083⋊C₃, #19096⋊C₃, …
maximal normal subgroup of	C₂×#1658⋊C₃, #19083⋊C₃, #19096⋊C₃, …
maximal quotient of	#7744⋊C₃, #7832⋊C₃, #8509⋊C₃, …

#1658⋊C₃ #68016

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

#1658⋊C3 #68016

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

#1658⋊C₃ #68016