C₇×#1846 #11730315

order	896
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#1846

Subgroups

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

248 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 896

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

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

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13240974, #13240984, #13240996, …
maximal subgroup of	C₇×#7271, C₇×#7275, C₇×#7279, …
maximal normal subgroup of	C₇×#7271, C₇×#7275, C₇×#7279, …
maximal quotient of	C₇×#8807, C₇×#8811, C₇×#8814, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5081);