#4829

order	192
automorphism	Aut(G) = C₂⁵×D₃×D₄ (order 1536) · Inn(G) = C₂×D₁₂ · 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×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#479
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#79025, #79029, #79030, …
maximal normal subgroup of	#79025, #79029, #79030, …
maximal quotient of	#68621, #68623, #68634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 388);