C₇×#12591 #13189807

order	1792
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#12591

Subgroups

	subgroup	G/H
center	C₂²×C₇×C₈	C₂³
commutator	C₂²	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₈ ◁ G

390 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 35
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3,4,5,6,7)(12,13)(14,17)(15,16)(18,19)(20,21,22,26,23,27,28,33,24,25,29,31,30,32,34,35);(8,9,10,11)(13,16)(17,19);(12,14,15,18)(13,17,16,19)(21,25)(26,31)(27,32)(33,35)

Conjugacy classes & character table

784 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5579);