#82451

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2295

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂³	C₂³×D₃
Fitting	C₂×C₃×C₄×Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₃
hypocenter	C₃	#2295
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10685821, #10685831, #10685841, …
maximal normal subgroup of	#10685821, #10685831, #10685841, …
maximal quotient of	#10716296, #10716316, C₄×#4980, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15027);