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C₄×#4428 #10625403

order	756
isoclinism	isoclinic class of #4427
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₄×C₇×C₉ ⋊ C₃
Hall	C₄×C₇ ⋊ 3₋¹⁺²
Sylow	C₇ ⋊ C₄×3₋¹⁺²

Subgroups

	subgroup	G/H
center	C₃×C₄	C₃×ΓL(1,𝔽₈)
commutator	C₃×C₇	C₃²×C₄
Frattini	C₂×C₃	C₂×C₃×ΓL(1,𝔽₈)
Fitting	C₄×C₇×C₉	C₃
socle, cosocle	C₂×C₃×C₇	C₂×C₃²
prime radical	C₂×C₃×ΓL(1,𝔽₈)	C₂×C₃
hypercenter	C₄×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₄×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₄×C₉

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 252

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

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13094146, #13094152, #13094157, …
maximal normal subgroup of	#13094146, #13094152, #13094157, …
maximal quotient of	C₈×#4428, C₂×C₄×#4428

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 34);