Finite Group Databasebeta

← Prev (#5572) Next (#5574) →

C₄×#239 #5573

order	192
automorphism	Aut(G) order 9216
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

Fitting	C₃×C₄×Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₄×#41

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂³×C₄
Frattini	C₂²	C₂³×D₃
Fitting	C₃×C₄×Q₈	C₂
socle, cosocle	C₂²×C₃	C₂⁴
prime radical	C₂×#239	C₂
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	C₄×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

(1,2,3)(8,10,11,14)(9,12,13,15);(8,9,11,13)(10,15,14,12);(2,3)(10,14)(12,15);(4,5,6,7)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68229, #68230, #68237, …
maximal normal subgroup of	#68229, #68230, #68237, …
maximal quotient of	C₄×#892, C₄×#893, C₄×#898, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1132);