C₇×#3093 #11742402

order	896
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#3093

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₇×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

414 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 31 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13386610, #13386618, #13386626, …
maximal subgroup of	C₇×#24020, C₇×#24021, C₇×#24022, …
maximal normal subgroup of	C₇×#24020, C₇×#24021, C₇×#24022, …
maximal quotient of	C₇×#20701, C₇×#20845, C₇×#20910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 17168);