#88530

order	432
isoclinism	isoclinic class of #88528
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

H(3,𝔽₃) ⋊ Q₁₆

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₄⋊(C₃⋊D₃)
commutator, cosocle	C₄×H(3,𝔽₃)	C₂²
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	C₈×H(3,𝔽₃)	C₂
prime radical	C₂×H(3,𝔽₃)	D₄
hypercenter	C₃×C₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	Q₁₆
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈

99 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

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

Polycyclic presentation:

2^4,3^3|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^2,f^e=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720255, #11720256, #11720513, …
maximal normal subgroup of	#11720255, #11720256, #11720513, …
maximal quotient of	#11720549, #11720551, C₂×#88530, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 177);