#80787

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#468 ⋊ C₂
Sylow	C₃ ⋊ #3175

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#468	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5038	C₂
hypercenter	#468	D₃
hypocenter	C₃	#3175
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

362 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)(4,6,7,10)(5,9,8,11)(13,18)(17,25)(19,24)(23,27);(2,3)(12,14,21,26,16,22,15,20)(13,17,24,27,18,25,19,23);(5,8)(9,11)(12,13)(14,17)(15,19)(16,18)(20,23)(21,24)(22,25)(26,27);(4,5)(6,9)(7,8)(10,11)(14,20)(15,21)(17,23)(19,24)(22,26)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3175, C₂×#80787, #11665035, …
maximal normal subgroup of	D₃×#3175, C₂×#80787, #11665035, …
maximal quotient of	#10795754, #10799456, #10799464, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 13363);