#79700

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×Q₁₆ ⋊ C₂
Sylow	C₃ ⋊ #3364

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×Q₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×D₁₂	C₂³
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#3364
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

336 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)(4,6,7,11)(5,9,8,10);(12,15,17,23)(13,19,20,27)(14,21,16,25)(18,26,24,22);(4,5,7,8)(6,10,11,9)(12,14,17,16)(13,18,20,24)(15,21,23,25)(19,26,27,22);(2,3)(5,8)(9,10)(12,13)(14,18)(15,22)(16,24)(17,20)(19,21)(23,26)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3364, C₂×#79700, #11674824, …
maximal normal subgroup of	D₃×#3364, C₂×#79700, #11674824, …
maximal quotient of	#10811802, #10811984, #10814133, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12276);