C₇×#1461 #11729930

order	896
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#1461

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

248 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 3 generators on 31 points:

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

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13240799, #13241016, #13244362, …
maximal subgroup of	C₇×#7206, C₇×#7286, C₇×C₈×#101, …
maximal normal subgroup of	C₇×#7206, C₇×#7286, C₇×C₈×#101, …
maximal quotient of	C₇×#7568, C₇×#7570, C₇×#7585, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4696);