C₇×#7534 #13184750

order	1792
nilpotency class	6
isoclinism	isoclinic class of #7533
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#7534

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1409
commutator	C₄×C₈	C₂×C₄×C₇
Frattini, cosocle	#494	C₂²×C₇
prime radical	C₂²×C₇	#353
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₄²×C₇ ◁ C₇×#494 ◁ G

128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 2 generators on 39 points:

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

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 522);