C₇×#2582 #11741891

order	896
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2582

Subgroups

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

484 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 896

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

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

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13256105, #13256132, #13256150, …
maximal subgroup of	C₇×#10230, C₇×#10234, C₇×#10237, …
maximal normal subgroup of	C₇×#10230, C₇×#10234, C₇×#10237, …
maximal quotient of	C₇×#13926, C₇×#14017, C₇×#14041, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16657);