C₂⁴×C₂₇⋊C₃ #13048207

order	1296
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃×C₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂⁴×C₂₇⋊C₃

Subgroups

	subgroup	G/H
center	C₂⁴×C₉	C₃²
commutator	C₃	C₂⁴×C₃×C₉
Frattini, cosocle	C₉	C₂⁴×C₃²
socle	C₂⁴×C₃	C₃×C₉
prime radical	C₂⁴×C₃²	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2)(9,10,13,11,15,21,17,24,29,12,16,20,18,25,28,26,32,34,19,14,22,27,23,30,33,31,35);(3,4)(10,14,16)(13,20,22)(15,23,25)(21,28,30)(24,31,32)(29,34,35);(5,6);(7,8)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂⁴×D₂₇⋊C₃, #72276267
maximal subgroup of	#120×C₂₇⋊C₃, C₂⁴×D₂₇⋊C₃, #72276267
maximal normal subgroup of	#120×C₂₇⋊C₃, C₂⁴×D₂₇⋊C₃, #72276267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 1925);

C24×C27⋊C3 #13048207