C₇×#1347 #11725713

order	896
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#1347

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#102
commutator	C₂×C₈	C₂×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂²×C₇
prime radical	C₇×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ G

126 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 44 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 — 2 generators on 31 points:

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

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1347, #13264021, #13264026, …
maximal subgroup of	C₂×C₇×#1347, C₇×#11703, C₇×#11704, …
maximal normal subgroup of	C₂×C₇×#1347, C₇×#11703, C₇×#11704, …
maximal quotient of	C₇×#7076, C₇×#7079, C₇×#7088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 479);