C₇×#2063 #11730532

order	896
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2063

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

228 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 27 points:

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#2063, #13382340, #13382349, …
maximal subgroup of	C₂×C₇×#2063, C₇×#23601, C₇×#23602, …
maximal normal subgroup of	C₂×C₇×#2063, C₇×#23601, C₇×#23602, …
maximal quotient of	C₇×#10587, C₇×#10589, C₇×#10591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5298);