C₇×#12410 #13189626

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12410

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×Q₈ ◁ G

276 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 1792

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

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

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, 5398);