C₇×2₋¹⁺⁴⋊C₃ #10621609

order	672
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×2₋¹⁺⁴⋊C₃
unambiguous semidirect product	2₋¹⁺⁴⋊(C₃×C₇)

Fitting, Hall	C₇×2₋¹⁺⁴ ⋊ C₃
Sylow	2₋¹⁺⁴ ⋊ C₃×C₇

	subgroup	G/H
center, socle	C₂×C₇	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃×C₇
Frattini	C₂	C₂²×C₇×A₄
Fitting	C₇×2₋¹⁺⁴	C₃
hypercenter	C₇×D₄	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₇ ◁ C₇×D₄

92 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 1191);