C₇×#1531 #11730000

order	896
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#1531

Subgroups

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

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

normal subgroups only

This group has 212 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) = 448

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

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13245684, #13245691, #13245940, …
maximal subgroup of	C₇×#8417, C₇×#8418, C₇×#8457, …
maximal normal subgroup of	C₇×#8417, C₇×#8418, C₇×#8457, …
maximal quotient of	C₇×#7579, C₇×#7588, C₇×#7591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4766);