C₇×#7452 #13184668

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×#7452

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	#357
commutator	C₂×C₁₆	C₂×C₄×C₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₇
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

102 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 54 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)(16,17,20,25,35,32,44,45,23,27,40,39,24,30,37,43,21,26,19,33,36,42,47,46,41,31,38,34,22,18,28,29);(8,9,11,13)(10,15,14,12)(16,18,21,30)(17,24,26,22)(19,34,20,39)(23,42,41,32)(25,38,33,40)(27,35,31,36)(28,43,37,29)(44,45,47,46)

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 440);