#87551

order	384
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

2₋¹⁺⁶ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	C₂²⋊(C₂²⋊A₄)
commutator, Fitting, cosocle, hypocenter	2₋¹⁺⁶	C₃
LCS	G ▷ 2₋¹⁺⁶
UCS	C₁ ◁ C₂

1056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(2,7,3)(4,5,16)(8,19,28)(9,23,11)(10,20,13)(12,24,18)(14,15,17)(21,31,29)(22,32,26)(25,30,27);(2,10)(3,13)(8,22)(9,25)(11,27)(12,29)(21,24)(26,28);(1,14,6,16)(2,21,10,24)(3,26,13,28)(4,17,15,5)(7,30,20,23)(8,25,22,9)(11,29,27,12)(18,32,31,19);(1,7)(2,3)(4,18)(5,19)(6,20)(8,11)(9,12)(10,13)(14,30)(15,31)(16,23)(17,32)(21,26)(22,27)(24,28)(25,29)

Polycyclic presentation:

3,2^7|d^2=h,e^2=h,b^a=bc,c^a=b,d^a=e,e^a=deh,f^a=fg,g^a=f,g^b=gh,f^c=fh,e^d=eh

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11715848, #11715849, #72265541, …
maximal subgroup of	#11710348, #11710353, C₂×#87551, …
maximal normal subgroup of	#11710348, #11710353, C₂×#87551, …
maximal quotient of	#11710743, #62871⋊C₃, #11710758, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20127);