C₇×#12778 #13189994

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#12778

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1790
commutator	C₂×Q₈	C₂²×C₄×C₇
Frattini, cosocle	C₂×#41	C₂³×C₇
prime radical	C₇×#3549	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

860 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(8,9)(10,18,20,22)(11,15,21,14)(12,16)(13,17)(19,23);(9,14)(11,21)(12,19)(15,17)(16,18)(22,23);(8,10)(9,14)(11,19)(12,21)(13,20)(15,17)(16,22)(18,23)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5766);