C₇×#11536 #13188752

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11536

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

376 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 448

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

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

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, 4524);