C₇×#2760 #11742069

order	896
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#2760

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₇×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

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

normal subgroups only

This group has 520 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)(9,12)(10,14)(16,17,20,18)(19,22,23,21)(24,25)(26,30)(27,28)(29,31);(9,12)(13,15)(17,18)(19,23)(25,28)(26,31);(8,9,11,12)(10,13,14,15)(16,18,20,17)(19,22,23,21)(24,26)(25,29)(27,31)(28,30);(8,10)(9,13)(11,14)(12,15)(16,19)(17,21)(18,22)(20,23)(24,25)(26,29)(27,28)(30,31)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13266409, #13266893, #13562118, …
maximal subgroup of	C₇×#12187, C₇×#12263, C₇×#36776, …
maximal normal subgroup of	C₇×#12187, C₇×#12263, C₇×#36776, …
maximal quotient of	C₇×#14375, C₇×#14508, C₇×#14572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16835);