C₇×#12864 #13190080

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12864

Subgroups

	subgroup	G/H
center	C₄×C₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×#41	C₂³×C₇
socle	C₂×C₇	#1820
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×#41 ◁ G

342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5852);