#564⋊C₃ #5466

order	192
isoclinism	isoclinic class of #564⋊C₃
automorphism	Aut(G) order 61440
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

unambiguous semidirect product

#564⋊C₃

Splittings

Fitting, derived, Sylow

#564 ⋊ C₃

Subgroups

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

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

3,2^6|b^2=fg,c^2=f,d^2=fg,e^2=f,b^a=c,c^a=bcg,d^a=e,e^a=deg,f^a=g,g^a=fg,d^b=df,e^b=efg,d^c=dfg,e^c=eg

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉
size	1	3	12	12	12	12	12	64	64
order	1	2	4	4	4	4	4	3	3
χ₁	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	3	3	-1	-1	-1	-1	0	0
χ₅	3	3	-1	3	-1	-1	-1	0	0
χ₆	3	3	-1	-1	3	-1	-1	0	0
χ₇	3	3	-1	-1	-1	3	-1	0	0
χ₈	3	3	-1	-1	-1	-1	3	0	0
χ₉	12	-4	0	0	0	0	0	0	0

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

Relationships

derived subgroup of	#11710972
maximal subgroup of	#1658⋊C₃, C₂×#564⋊C₃, #3556⋊C₃, …
maximal normal subgroup of	#1658⋊C₃, C₂×#564⋊C₃, #3556⋊C₃, …
maximal quotient of	#2810⋊C₃, #2845⋊C₃, C₂×#564⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1025);

#564⋊C3 #5466