C₂²×C₇×C₂²⋊Q₈ #11744385

order	896
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₇ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×C₇×C₂²⋊Q₈

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₇
prime radical	C₂⁵×C₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

1840 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 5 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 19151);