#69791

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1837

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1837
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2^7,3|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682230, #10682231, #10682233, …
maximal normal subgroup of	#10682230, #10682231, #10682233, …
maximal quotient of	#10688848, #10688851, #10688864, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 2367);