C₇×#13684 #13190900

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13684

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₇×#3293	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×C₄⋊Q₈ ◁ G

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 224

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

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6672);