C₇×#2169 #11730638

order	896
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2169

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂³
commutator	C₂²	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₇
socle	C₂²×C₇	C₂²×C₈
prime radical	C₂×C₇×D₄	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

162 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 448

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13241237, #13265504, #13265511, …
maximal subgroup of	C₇×#7380, C₇×#12038, C₇×#12039, …
maximal normal subgroup of	C₇×#7380, C₇×#12038, C₇×#12039, …
maximal quotient of	C₇×#12039, C₇×#12051, C₇×#12054, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5404);