C₇×#2812 #11742121

order	896
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₇×#2812

Subgroups

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

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

normal subgroups only

This group has 712 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) = 27
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13265843, #13266531, D₇×#2812, …
maximal subgroup of	C₇×#12093, C₇×#12205, C₂×C₇×#2812, …
maximal normal subgroup of	C₇×#12093, C₇×#12205, C₂×C₇×#2812, …
maximal quotient of	C₇×#14621, C₇×#14622, C₇×#14627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16887);