C₇×#7393 #13184609

order	1792
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×#7393

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₇×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂²×C₇
socle, prime radical	C₂²×C₇	C₂²⋊C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

110 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 1792

A minimal faithful permutation representation — 2 generators on 47 points:

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

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, 381);