C₇×#2240 #11730709

order	896
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#2240

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
prime radical	C₇×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ G

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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)(8,9)(10,13)(11,12)(14,15)(17,23)(18,28)(19,30)(20,31)(21,24)(25,26)(27,29);(9,12)(13,15)(16,17,20,25,21,26,19,23,22,27,30,18,24,28,31,29);(8,10,11,14)(9,13,12,15)(16,18,19,17,24,23,20,28,22,25,31,27,21,29,30,26)

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13267385, #13267392, #13267399, …
maximal subgroup of	C₇×#12340, C₇×#12341, C₇×#12342, …
maximal normal subgroup of	C₇×#12340, C₇×#12341, C₇×#12342, …
maximal quotient of	C₇×#12888, C₇×#13147, C₇×#13202, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5475);