C₇×#2843 #11742152

order	896
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#2843

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

348 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 31
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 4 generators on 31 points:

(1,2,3,4,5,6,7)(8,9,11,13)(10,12,14,15)(16,17)(18,22)(19,23)(20,21);(10,14)(12,15)(17,21)(19,22)(24,25,27,29)(26,30,31,28);(8,9)(10,12)(11,13)(14,15)(16,18)(17,22)(19,21)(20,23)(24,26,27,31)(25,28,29,30);(8,10)(9,12)(11,14)(13,15)(16,19,20,22)(17,23,21,18)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13266385, #13266869, D₇×#2843, …
maximal subgroup of	C₇×#12183, C₇×#12259, C₂×C₇×#2843, …
maximal normal subgroup of	C₇×#12183, C₇×#12259, C₂×C₇×#2843, …
maximal quotient of	C₇×#14638, C₇×#14639, C₇×#14640, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16918);