C₇×#2571 #11741880

order	896
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2571

Subgroups

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

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

normal subgroups only

This group has 380 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) = 31
transitive degree	μ_t(G) = 896

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

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13256820, #13256825, #13256830, …
maximal subgroup of	C₇×#10351, C₇×#10352, C₇×#10353, …
maximal normal subgroup of	C₇×#10351, C₇×#10352, C₇×#10353, …
maximal quotient of	C₇×#13911, C₇×#13982, C₇×#14030, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16646);