#79789

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#440 ⋊ C₂
Sylow	C₃ ⋊ #3126

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#440	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5562	C₂
hypercenter	#440	D₃
hypocenter	C₃	#3126
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

382 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)(13,17)(18,24)(19,25)(26,27);(12,14,15,21)(13,18,17,24)(16,23,22,20)(19,26,25,27);(2,3)(5,8)(9,11)(12,13)(14,18)(15,17)(16,19)(20,27)(21,24)(22,25)(23,26);(2,3)(4,5)(6,9)(7,8)(10,11)(13,17)(14,20)(16,22)(18,26)(21,23)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3126, C₂×#79789, #11663556, …
maximal normal subgroup of	D₃×#3126, C₂×#79789, #11663556, …
maximal quotient of	#10795286, #10798306, #10798312, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12365);