C₇×#2801 #11742110

order	896
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#2801

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

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

normal subgroups only

This group has 420 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)(16,17,19,21)(18,22,23,20)(25,26)(29,30);(9,12)(13,15)(24,25,28,26)(27,30,31,29);(8,9,11,12)(10,13,14,15)(16,17)(18,20)(19,21)(22,23)(24,26,28,25)(27,30,31,29);(8,10,11,14)(9,13,12,15)(16,18,19,23)(17,20,21,22)(24,27,28,31)(25,29,26,30)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13561820, #13561925, #13563572, …
maximal subgroup of	C₇×#36766, C₇×#36769, C₇×#36826, …
maximal normal subgroup of	C₇×#36766, C₇×#36769, C₇×#36826, …
maximal quotient of	C₇×#14434, C₇×#14459, C₇×#14504, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16876);