Finite Group Databasebeta

← Prev (#5796) Next (#5798) →

C₂×#936 #5797

order	192
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

Splittings

C₃ ⋊ C₂×#124

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting, prime radical	C₂²×C₃×D₄	C₂
socle	C₂³×C₃	C₂³
hypercenter	C₂²×D₄	D₃
hypocenter	C₃	C₂×#124
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 17 points:

(1,2)(3,4,5)(7,9)(10,12)(11,14)(13,16)(15,17);(10,11,13,15)(12,17,16,14);(4,5)(6,7,8,9)(12,16)(14,17);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#4473, #70065, #70069, …
maximal normal subgroup of	C₂×#4473, #70065, #70069, …
maximal quotient of	C₂×#4945, C₂×#4947, C₂×#4953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1356);