C₇×#12366 #13189582

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12366
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12366

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3292	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

364 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,13,15,23)(10,17)(11,20)(14,19,16,22)(18,21);(10,18)(11,20)(13,19)(14,16)(17,21)(22,23);(8,9)(10,19)(11,14)(12,15)(13,18)(16,20)(17,23)(21,22)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#12366, #72259974
maximal subgroup of	D₇×#12366, C₇×#489347, #72259974
maximal normal subgroup of	D₇×#12366, C₇×#489347, #72259974

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5354);