C₇×#1533 #11730002

order	896
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#1533

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13241524, #13241566, #13246313, …
maximal subgroup of	C₇×#7481, C₇×#7495, C₇×#8522, …
maximal normal subgroup of	C₇×#7481, C₇×#7495, C₇×#8522, …
maximal quotient of	C₇×#7592, C₇×#7595, C₇×#7601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4768);