C₇×#536 #90557

order	448
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#536

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂×C₇×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

270 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,23,24,25,26,27);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(2,4)(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(6,12)(10,13)(11,20)(18,19)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11729617, #11729618, #11729619, …
maximal subgroup of	#11729617, #11729618, #11729619, …
maximal normal subgroup of	#11729617, #11729618, #11729619, …
maximal quotient of	C₇×#2310, C₇×#2311, C₇×#2317, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1319);