C₇×#1306 #11725672

order	896
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#1306

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#95
commutator	C₂³	C₄²×C₇
Frattini, cosocle	C₂³×C₄	C₂²×C₇
prime radical	C₂⁴×C₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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)(11,13)(12,15)(16,17)(18,24,25,21)(19,23)(20,22)(26,30,31,28)(27,29);(8,10)(9,12,13,14)(11,15)(16,18,19,26,20,25,27,31)(17,21,23,28,22,24,29,30)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13241281, #13241377, D₇×#1306, …
maximal subgroup of	C₇×#7395, C₇×#7427, C₂×C₇×#1306, …
maximal normal subgroup of	C₇×#7395, C₇×#7427, C₂×C₇×#1306, …
maximal quotient of	C₇×#7032, C₇×#7033, C₇×#7037, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 438);