#79992

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #3118

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting, prime radical	C₃×#447	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	#447	D₃
hypocenter	C₃	#3118
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3118, C₂×#79992, #11663547, …
maximal normal subgroup of	D₃×#3118, C₂×#79992, #11663547, …
maximal quotient of	#10795272, #10798256, #10798275, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12568);