C₇×#2640 #11741949

order	896
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₇×#2640

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

500 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 448

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

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13255152, #13255166, #13255208, …
maximal subgroup of	C₇×#10090, C₇×#10092, C₇×#10098, …
maximal normal subgroup of	C₇×#10090, C₇×#10092, C₇×#10098, …
maximal quotient of	C₇×#14260, C₇×#14261, C₇×#14273, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16715);