C₇×#1702 #11730171

order	896
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#1702

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

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

normal subgroups only

This group has 162 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) = 448

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

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

Conjugacy classes & character table

161 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1702, #13336475, #13336504, …
maximal subgroup of	C₂×C₇×#1702, C₇×#19155, C₇×#19157, …
maximal normal subgroup of	C₂×C₇×#1702, C₇×#19155, C₇×#19157, …
maximal quotient of	C₇×#8657, C₇×#9226, C₇×#9230, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4937);