C₇×#12412 #13189628

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

Aliases

direct product

C₇×#12412

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×#446 ◁ G

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

normal subgroups only

This group has 280 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)(8,9,14,18)(10,19,11,22)(12,17,13,16)(15,21,23,20)(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);(8,10,14,11)(9,15,18,23)(12,21,13,20)(16,22,17,19)(24,26,30,38)(25,31,34,48)(27,36,29,45)(28,41,43,55)(32,47,44,49)(33,50,52,54)(35,51,53,40)(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)(28,42,43,39)(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, 5400);