C₇×#559 #90580

order	448
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₇×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

236 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1342);