Finite Group Databasebeta

← Prev (#66534) Next (#66536) →

C₃×C₇⋊C₁₆ #66535

order	336
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂²×C₄×AGL(1,𝔽₇) (order 672) · Inn(G) = D₇ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₃×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₇⋊C₁₆

Splittings

derived, Sylow	C₇ ⋊ C₃×C₁₆
Hall	C₃×C₇ ⋊ C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈	D₇
commutator, hypocenter	C₇	C₃×C₁₆
Frattini	C₈	C₃×D₇
Fitting	C₃×C₇×C₈	C₂
socle, prime radical	C₂×C₃×C₇	C₈
cosocle	C₇×C₈	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₈

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

(2,3)(4,5)(6,7)(8,13,16,15,18,23,10,14,12,20,22,9,11,19,17,21)(24,25,26);(1,2,4,6,7,5,3)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₇⋊C₁₆, D₂₁⋊C₁₆, #10620470, …
maximal normal subgroup of	D₃×C₇⋊C₁₆, D₂₁⋊C₁₆, #10620470, …
maximal quotient of	C₃×C₇⋊C₃₂, C₂×C₃×C₇⋊C₁₆, C₉×C₇⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 4);