C₂×C₇×C₂²⋊Q₈ #90544

order	448
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×C₇×C₂²⋊Q₈

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

322 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1306);