#78542

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2960

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#405	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#405	D₃
hypocenter	C₃	#2960
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

364 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10704693, #10713712, #10713716, …
maximal normal subgroup of	#10704693, #10713712, #10713716, …
maximal quotient of	#10793136, #10793140, #10793155, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11118);