C₇×#23925 #13201141

order	1792
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₇×#23925

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₇×#41 ◁ G

2080 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 2080 conjugacy classes of subgroups — too many to draw. Only its 302 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 — 4 generators on 31 points:

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

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 16913);