C₇×#1500 #11729969

order	896
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#1500

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₄
prime radical	C₇×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,12,13,14)(16,17)(18,24,26,21)(19,22)(20,23)(25,30,31,28)(27,29);(8,9)(10,14)(11,13)(12,15)(16,18)(17,21)(19,25)(20,26)(22,28)(23,24)(27,31)(29,30);(16,19,20,27)(17,22,23,29)(18,25,26,31)(21,28,24,30)

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13244836, #13244843, #13244850, …
maximal subgroup of	C₇×#8247, C₇×#8248, C₇×#8249, …
maximal normal subgroup of	C₇×#8247, C₇×#8248, C₇×#8249, …
maximal quotient of	C₇×#7560, C₇×#7562, C₇×#7567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4735);