C₇×#18732 #13195948

order	1792
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#18732

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂³×C₇	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

808 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 55 points:

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

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, 11720);