C₇×#12083 #13189299

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12083

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle, prime radical	C₂²×C₇	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₇×C₈² ◁ G

210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 47
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,12,10,11,13,15,14)(16,17,20,24)(18,26,28,45)(19,30,33,44)(21,35,32,25)(22,37,38,27)(23,40,42,34)(29,46,36,41)(31,43,47,39);(9,13)(10,14)(16,18,20,28)(17,22,24,38)(19,31,33,47)(21,36,32,29)(23,41,42,46)(25,43,35,39)(26,44,45,30)(27,40,37,34);(8,10,12,13,11,14,15,9)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5071);