C₇×#1998 #11730467

order	896
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#1998

Subgroups

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

228 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 112 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)(10,14)(11,12)(13,15)(16,17,20,23,19,22,24,30)(18,21,26,29,25,28,27,31);(9,12)(13,14)(16,18,24,26,19,25,20,27)(17,21,23,31,22,28,30,29);(8,10,11,15)(9,13,12,14)(17,22)(21,28)(23,30)(29,31)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13246107, #13246114, #13257761, …
maximal subgroup of	C₇×#8485, C₇×#8486, C₇×#10497, …
maximal normal subgroup of	C₇×#8485, C₇×#8486, C₇×#10497, …
maximal quotient of	C₇×#9880, C₇×#9881, C₇×#9892, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5233);