C₃₇×C₂₇⋊C₃ #15688490

order	2997
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×C₂₇⋊C₃
unambiguous semidirect product	C₂₇⋊(C₃×C₃₇)

Subgroups

	subgroup	G/H
center	C₉×C₃₇	C₃²
commutator	C₃	C₃×C₉×C₃₇
Frattini, cosocle	C₉	C₃²×C₃₇
socle	C₃×C₃₇	C₃×C₉
prime radical	C₃²×C₃₇	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₃₇ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 999

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39,42,40,44,50,46,53,58,41,45,49,47,54,57,55,61,63,48,43,51,56,52,59,62,60,64);(39,43,45)(42,49,51)(44,52,54)(50,57,59)(53,60,61)(58,63,64)

Conjugacy classes & character table

1221 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of

#41008316, #41008320, #41008323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2997, 6);

C37×C27⋊C3 #15688490