C₇×#8242 #13185458

order	1792
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#8242

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	C₄×C₂²⋊C₄
prime radical	C₂³×C₇	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

384 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 39
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 1230);