C₇×#12593 #13189809

order	1792
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇×C₈ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₇×#12593

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₄²
commutator	C₄	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×C₇	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

338 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 222 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)(10,13)(11,12)(14,15)(16,17,21,27,22,29,37,44,23,28,36,43,20,26,19,25)(18,30,33,46,34,24,31,40,35,42,38,39,32,41,47,45);(9,12)(13,15)(16,18)(17,24,28,41)(19,31)(20,32)(21,38)(22,34)(23,35)(25,39,44,46)(26,30,29,42)(27,45,43,40)(33,36)(37,47);(8,10,11,14)(9,13,12,15)(16,19,20,36,23,37,22,21)(17,25,26,43,28,44,29,27)(18,31,32,33,35,47,34,38)(24,39,30,40,41,46,42,45)

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5581);