C₇×#12388 #13189604

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12388

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3403	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#446 ◁ G

296 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 78 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)(28,40,43,51)(31,46,48,37)(33,39,52,42)(36,54,45,50)(41,47,55,49);(11,20)(14,19)(16,23)(18,21)(24,26)(25,31)(27,36)(28,41)(29,45)(30,38)(32,47)(33,50)(34,48)(35,40)(37,42)(39,46)(43,55)(44,49)(51,53)(52,54);(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)(28,42,43,39)(31,47,48,49)(33,51,52,40)(35,50,53,54)(37,55,46,41)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5376);