C₇×#2785 #11742094

order	896
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₄³

Aliases

direct product

C₇×#2785

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

428 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 176 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 31 points:

(1,2,3,4,5,6,7)(8,9,12,10)(11,14,15,13)(16,17)(18,21)(19,20)(22,23)(25,26)(29,30);(9,10)(11,15)(17,19)(21,22)(24,25,28,26)(27,30,31,29);(8,10,12,9)(11,14,15,13)(16,18,20,23)(17,21,19,22)(24,26,28,25)(27,30,31,29);(8,11,12,15)(9,13,10,14)(16,19,20,17)(18,22,23,21)(24,27)(25,29)(26,30)(28,31)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13266155, #13266741, #13562752, …
maximal subgroup of	C₇×#12145, C₇×#12239, C₇×#36796, …
maximal normal subgroup of	C₇×#12145, C₇×#12239, C₇×#36796, …
maximal quotient of	C₇×#14399, C₇×#14476, C₇×#14589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16860);