#81975

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#501 ⋊ C₂
Sylow	C₃ ⋊ #3164

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#501	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
hypercenter	#501	D₃
hypocenter	C₃	#3164
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 27 points:

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

Polycyclic presentation:

2^7,3|a^2=e,b^2=e,c^2=fg,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,h^c=h^2,h^d=h^2

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3164, C₂×#81975, #11664822, …
maximal normal subgroup of	D₃×#3164, C₂×#81975, #11664822, …
maximal quotient of	#10795636, #10798551, #10798565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14551);