C₇×#12296 #13189512

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12296

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#424 ◁ G

278 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 74 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)(10,13,17,20)(11,14,18,21)(16,22,23,19)(25,31)(26,35)(27,38)(28,29)(32,46)(33,50)(34,45)(36,52)(37,42)(39,43)(40,41)(44,47)(48,55)(49,53);(8,10)(9,13)(11,16)(12,17)(14,19)(15,20)(18,23)(21,22)(24,25,30,34)(26,32,37,49)(27,39,41,50)(28,33,29,43)(31,40,45,38)(35,47,42,48)(36,53,52,46)(44,54,55,51);(8,11,12,18)(9,14,15,21)(10,16,17,23)(13,19,20,22)(24,26)(25,32)(27,36)(28,42)(29,35)(30,37)(31,44)(33,48)(34,49)(38,51)(39,53)(40,54)(41,52)(43,47)(45,55)(46,50)

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, 5284);