C₇×#11953 #13189169

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11953

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

296 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 55
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9,12,17,13,16,11,15)(10,18,19,23,20,14,21,22)(24,25,28,34,42,50,43,52,30,32,27,33,29,35,45,54)(26,36,38,49,46,53,41,31,40,48,37,55,39,51,44,47);(8,10)(9,14)(11,19)(12,21)(13,20)(15,22)(16,18)(17,23)(24,26,30,40)(25,31,32,47)(27,41,28,44)(29,46,42,39)(33,51,34,53)(35,55,50,49)(36,54,48,52)(37,45,38,43);(8,11,13,12)(9,15,16,17)(10,19,20,21)(14,22,18,23)(25,32)(31,47)(33,34)(35,50)(36,48)(49,55)(51,53)(52,54)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4941);