C₇×#12045 #13189261

order	1792
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12045

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₄×D₈
prime radical	C₂×C₇×D₁₆	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₇×C₈² ◁ G

254 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5033);