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C₇×C₂²⋊C₉ #6971

order	252
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₂×C₃²×S₄ (order 432) · Inn(G) = A₄ · Out(G) = C₂²×C₃²
homology	Abelianization (H₁) = C₇×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×C₂²⋊C₉
unambiguous semidirect product	C₂²⋊(C₇×C₉)

Splittings

derived, Sylow	C₂² ⋊ C₇×C₉
Hall	C₂²×C₇ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇	A₄
commutator, hypocenter	C₂²	C₇×C₉
Frattini	C₃	C₇×A₄
Fitting, socle, prime radical	C₂²×C₃×C₇	C₃
cosocle	C₂²×C₃	C₃×C₇
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₇⋊((C₂²⋊C₉)⋊C₂), C₇⋊((C₂²⋊C₉)⋊C₄), D₇×(C₂²⋊C₉)⋊C₂, …
maximal subgroup of	C₇×(C₂²⋊C₉)⋊C₂, C₇⋊((C₂²⋊C₉)⋊C₂), D₇×C₂²⋊C₉, …
maximal normal subgroup of	C₇×(C₂²⋊C₉)⋊C₂, C₇⋊((C₂²⋊C₉)⋊C₂), D₇×C₂²⋊C₉, …
maximal quotient of	C₇×Q₈⋊C₉, C₂×C₇×C₂²⋊C₉, C₇×C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 10);