C₂⁴×D₇ #6564

order	224
isoclinism	isoclinic class of D₇
automorphism	Aut(G) order 13547520
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product	C₂⁴×D₇
unambiguous semidirect product	C₇⋊C₂⁵
unambiguous semidirect product	(C₂×C₇)⋊C₂⁴

Fitting, socle	C₂⁴×C₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂⁵

	subgroup	G/H
center, hypercenter	C₂⁴	D₇
commutator, cosocle, hypocenter	C₇	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁴×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁴

748 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 5 generators on 15 points:

(1,2)(3,4)(5,6)(7,8)(10,11)(12,13)(14,15);(7,8)(9,10,12,14,15,13,11);(5,6);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 196);