C₇×#32877 #13210093

order	1792
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#32877

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ G

594 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 39 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 25865);