C₇×#11540 #13188756

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11540

Subgroups

	subgroup	G/H
center	C₄×C₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₇
socle	C₂×C₇	#1467
prime radical	C₂³×C₇	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₇×#405 ◁ G

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

normal subgroups only

This group has 262 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) = 39
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4528);