#5022

order	192
automorphism	Aut(G) = C₂⁵×D₃×D₄ (order 1536) · Inn(G) = C₂×#67 · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	#479
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

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

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

Polycyclic presentation:

2^6,3|a^2=e,b^2=ef,c^2=f,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,d^c=df

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#79620, #79630, #79633, …
maximal normal subgroup of	#79620, #79630, #79633, …
maximal quotient of	#69458, #69460, #69468, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 581);