#79669

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×SD₁₆ ⋊ C₂
Sylow	C₃ ⋊ #3179

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×SD₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₄×SD₁₆	D₃
hypocenter	C₃	#3179
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

374 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)(12,15,16,13)(14,22,21,18)(17,24,25,19)(20,23,27,26);(12,14,17,23,16,21,25,26)(13,18,19,20,15,22,24,27);(2,3)(5,8)(9,11)(13,15)(14,21)(19,24)(23,26);(4,5)(6,9)(7,8)(10,11)(12,13,16,15)(14,20,21,27)(17,24,25,19)(18,26,22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3179, C₂×#79669, #11665079, …
maximal normal subgroup of	D₃×#3179, C₂×#79669, #11665079, …
maximal quotient of	#10795764, #10799481, #10799495, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12245);