#69280

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ Q₈×C₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₁₆
Fitting	C₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₄⋊C₁₆	D₃
hypocenter	C₃	Q₈×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10706952, #10706956, #10706966, …
maximal normal subgroup of	#10706952, #10706956, #10706966, …
maximal quotient of	#10710528, #10710532, #10712799, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1856);