C₇×#1286 #11725652

order	896
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#1286

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#95
commutator	C₂×C₄	C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂²×C₇
prime radical	C₂³×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

156 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,13)(10,11)(14,15)(16,17)(18,24,26,30)(19,22)(20,23)(21,25,28,31)(27,29);(8,9,11,14,12,15,10,13)(16,18)(17,21,23,28)(19,25)(20,26)(22,24,29,30)(27,31)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1286, #13243574, #13243578, …
maximal subgroup of	C₂×C₇×#1286, C₇×#7956, C₇×#7957, …
maximal normal subgroup of	C₂×C₇×#1286, C₇×#7956, C₇×#7957, …
maximal quotient of	C₇×#7018, C₇×#7022, C₇×#7024, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 418);