C₅₃×H(5,𝔽₃) #27166253

order	12879
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₅₃ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₅₃×H(5,𝔽₃)

Subgroups

	subgroup	G/H
center, socle	C₃×C₅₃	C₃⁴
commutator, Frattini, cosocle	C₃	C₃⁴×C₅₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅₃ ◁ G

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1431

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,5)(3,9,20)(4,8,15)(6,16,11)(7,19,13)(10,22,26)(12,18,14)(17,27,23)(21,25,24)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80);(1,3,10)(2,6,17)(4,12,23)(5,14,25)(7,18,22)(8,9,21)(11,24,13)(15,16,26)(19,20,27);(3,11,12)(6,18,9)(10,23,24)(14,20,16)(17,21,22)(25,26,27);(2,7,8)(5,15,19)(6,18,9)(14,16,20)(17,22,21)(25,26,27)

Conjugacy classes & character table

4399 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₃×UT(4,𝔽₃), C₅₃×#10624748, C₅₃×#10624760, …
maximal normal subgroup of	C₅₃×UT(4,𝔽₃), C₅₃×#10624748, C₅₃×#10624760, …
maximal quotient of	C₅₃×#10624862, C₅₃×#10624880, C₅₃×#10624883, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12879, 65);

C53×H(5,𝔽3) #27166253