#4720

order	192
automorphism	Aut(G) = C₂⁷×D₃ (order 768) · Inn(G) = C₂²×D₃ · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #433

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₃×C₂²⋊C₈	C₂
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²⋊C₈	D₃
hypocenter	C₃	#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2^6,3|a^2=e,c^2=d,d^2=f,b^a=bf,g^a=g^2,c^b=ce

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#4720, #78584, #78604, …
maximal normal subgroup of	C₂×#4720, #78584, #78604, …
maximal quotient of	#68598, #68604, #68606, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 279);