C₇×#12385 #13189601

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

Aliases

direct product

C₇×#12385

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×Q₈ ◁ G

300 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 82 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)(8,9,12,15)(10,16,17,23)(11,19,20,14)(13,21,22,18)(24,25,30,34)(26,35,38,53)(27,32,29,44)(28,40,43,54)(31,46,49,37)(33,39,51,42)(36,52,45,50)(41,47,55,48);(11,20)(14,19)(16,23)(18,21)(24,26)(25,31)(27,36)(28,41)(29,45)(30,38)(32,47)(33,50)(34,49)(35,40)(37,42)(39,46)(43,55)(44,48)(51,52)(53,54);(8,10,12,17)(9,13,15,22)(11,21,20,18)(14,23,19,16)(24,27,30,29)(25,32,34,44)(26,36,38,45)(28,39,43,42)(31,47,49,48)(33,40,51,54)(35,52,53,50)(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, 5373);