#80791

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#468 ⋊ C₂
Sylow	C₃ ⋊ #3183

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	#5039	C₂
hypercenter	#468	D₃
hypocenter	C₃	#3183
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

370 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3183, C₂×#80791, #11665161, …
maximal normal subgroup of	D₃×#3183, C₂×#80791, #11665161, …
maximal quotient of	#10795787, #10799503, #10799514, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 13367);