C₇×#554 #90575

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

Aliases

direct product

C₇×#554

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂×C₇×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

228 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.

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

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11729677, #11729678, #11729679, …
maximal subgroup of	#11729677, #11729678, #11729679, …
maximal normal subgroup of	#11729677, #11729678, #11729679, …
maximal quotient of	C₇×#2345, C₇×D₄×C₄⋊C₄, C₇×#2349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1337);