C₇×#2815 #11742124

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×#2815

Subgroups

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

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

normal subgroups only

This group has 644 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) = 448

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

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13254428, #13254445, #13265849, …
maximal subgroup of	C₇×#9980, C₇×#9983, C₇×#12094, …
maximal normal subgroup of	C₇×#9980, C₇×#9983, C₇×#12094, …
maximal quotient of	C₇×#14621, C₇×#14623, C₇×#14630, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16890);