C₂⁵×D₇ #90633

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product	C₂⁵×D₇
unambiguous semidirect product	C₇⋊C₂⁶
unambiguous semidirect product	(C₂×C₇)⋊C₂⁵

Fitting, socle	C₂⁵×C₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂⁶

	subgroup	G/H
center, hypercenter	C₂⁵	D₇
commutator, cosocle, hypocenter	C₇	C₂⁶
Frattini	C₁	G
Fitting, socle	C₂⁵×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁵

5650 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 6 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1395);