#78508

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2962

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×M₁₆	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#2962
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

356 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)(14,20)(17,23)(21,26)(24,27);(12,14,15,21,16,20,22,26)(13,17,18,24,19,23,25,27);(4,6,7,11)(5,8,9,10);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10689667, #10689671, #10689681, …
maximal normal subgroup of	#10689667, #10689671, #10689681, …
maximal quotient of	#10793169, #10793186, #10793830, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11084);