C₇×#1834 #11730303

order	896
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#1834

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

248 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 3 generators on 27 points:

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

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13240832, #13240847, #13246857, …
maximal subgroup of	C₇×#7217, C₇×#7222, C₇×#8646, …
maximal normal subgroup of	C₇×#7217, C₇×#7222, C₇×#8646, …
maximal quotient of	C₇×#8784, C₇×#8787, C₇×#8790, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5069);