C₇×#12411 #13189627

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

Aliases

direct product

C₇×#12411

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂×C₇×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×Q₈ ◁ G

284 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 86 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)(8,9,14,18)(10,19,11,22)(12,17,13,16)(15,21,23,20)(24,25,30,34)(26,35,38,54)(27,32,29,44)(28,40,43,52)(31,46,49,37)(33,39,51,42)(36,53,45,50)(41,47,55,48);(8,10,14,11)(9,15,18,23)(12,21,13,20)(16,22,17,19)(24,26,30,38)(25,31,34,49)(27,36,29,45)(28,41,43,55)(32,47,44,48)(33,50,51,53)(35,52,54,40)(37,39,46,42);(8,11,14,10)(9,15,18,23)(12,20,13,21)(16,22,17,19)(24,27,30,29)(25,32,34,44)(26,36,38,45)(28,39,43,42)(31,47,49,48)(33,40,51,52)(35,53,54,50)(37,41,46,55)

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, 5399);