C₇×#2861 #11742170

order	896
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#2861

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

524 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 216 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)(9,11)(13,16)(17,19)(21,23)(25,26);(12,13)(14,17)(15,16)(18,19)(20,21,24,23)(22,25,27,26);(8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,22,24,27)(21,25,23,26);(12,14,15,18)(13,17,16,19)(20,23,24,21)(22,26,27,25)

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13242138, #13242230, #13242322, …
maximal subgroup of	C₇×#7656, C₇×#7676, C₇×#7696, …
maximal normal subgroup of	C₇×#7656, C₇×#7676, C₇×#7696, …
maximal quotient of	C₇×#14689, C₇×#14700, C₇×#14701, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16936);