C₇×#2005 #11730474

order	896
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2005

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂×C₇×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

362 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 92 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)(8,9)(10,11)(12,13)(14,22)(15,19)(16,25)(17,21)(18,23)(20,26)(24,27);(12,14,17,15)(13,18,21,27)(16,24,26,23)(19,25,22,20);(9,11)(12,15,17,14)(13,19,21,22)(16,23,26,24)(18,20,27,25)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#2005, #13380706, #13380722, …
maximal subgroup of	C₂×C₇×#2005, C₇×#23399, C₇×#23401, …
maximal normal subgroup of	C₂×C₇×#2005, C₇×#23399, C₇×#23401, …
maximal quotient of	C₇×#9917, C₇×#9919, C₇×#9925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5240);