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C₃×C₇⋊C₈ #4234

order	168
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂³×AGL(1,𝔽₇) (order 336) · Inn(G) = D₇ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₇⋊C₈

Splittings

derived, Sylow	C₇ ⋊ C₃×C₈
Hall	C₃×C₇ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	D₇
commutator, hypocenter	C₇	C₃×C₈
Frattini	C₄	C₃×D₇
Fitting	C₃×C₄×C₇	C₂
socle, prime radical	C₂×C₃×C₇	C₄
cosocle	C₄×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₄

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3)(4,5)(6,7)(8,12,14,9,11,15,10,13)(16,17,18);(1,2,4,6,7,5,3)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₇⋊C₈, D₂₁⋊C₈, #66558, …
maximal normal subgroup of	D₃×C₇⋊C₈, D₂₁⋊C₈, #66558, …
maximal quotient of	C₃×C₇⋊C₁₆, C₂×C₃×C₇⋊C₈, C₉×C₇⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 4);