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C₇×#157 #6992

order	252
isoclinism	isoclinic class of #157
automorphism	Aut(G) = C₂×C₃×AΓL(1,𝔽₉) (order 864) · Inn(G) = #157 · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₃

Aliases

Splittings

Fitting, socle, Hall	C₃²×C₇ ⋊ C₄
derived, Sylow	C₃² ⋊ C₄×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	#157
commutator, hypocenter	C₃²	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₄
prime radical	C₇×C₃⋊D₃	C₂
cosocle	C₃⋊D₃	C₂×C₇
LCS	G ▷ C₃²
UCS	C₁ ◁ C₇

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11759722, PSU(3,𝔽₂)⋊D₇, AGL(1,𝔽₉)⋊D₇, …
maximal subgroup of	C₇×AGL(1,𝔽₉), C₇⋊AGL(1,𝔽₉), D₇×#157, …
maximal normal subgroup of	C₇×AGL(1,𝔽₉), C₇⋊AGL(1,𝔽₉), D₇×#157, …
maximal quotient of	C₇×#622, C₂×C₇×#157, C₇×#1088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 31);