C₇×#9437 #13186653

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9437

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

556 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 126 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)(8,9,13,15,12,10,11,14)(17,21)(18,25)(19,22)(24,29)(26,27)(28,30);(8,10,12,9)(11,14,13,15)(16,17,22,29,20,24,19,21)(18,23,25,30,27,31,26,28);(11,13)(14,15)(16,18)(17,23)(19,26)(20,27)(21,28)(22,25)(24,31)(29,30)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2425);