#78469

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₄×C₈ ⋊ C₂
Sylow	C₃ ⋊ #3168

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5115	C₂
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#3168
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

380 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)(4,6,7,10)(5,9,8,11);(12,15,16,23,17,24,14,21)(13,19,18,22,20,26,25,27);(5,8)(9,11)(12,14,17,16)(13,18,20,25)(15,21,24,23)(19,22,26,27);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,18)(15,22)(16,25)(17,20)(19,23)(21,26)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10705169, #10705172, #10705174, …
maximal normal subgroup of	#10705169, #10705172, #10705174, …
maximal quotient of	#10795734, #10799396, #10799406, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11045);