C₅×C₂₇² #16813951

order	3645
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 1259712
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂₇

Aliases

abelian

C₅×C₂₇²

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉²	C₃²×C₅
socle, prime radical	C₃²×C₅	C₉²
LCS	G ▷ C₁
UCS	C₁ ◁ G

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 3645

A minimal faithful permutation representation — 2 generators on 59 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)(55,56,57,58,59);(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)

Conjugacy classes & character table

3645 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₅×C₂₇², C₅×C₂₇×C₈₁, C₃×C₅×C₂₇², …
maximal normal subgroup of	C₂×C₅×C₂₇², C₅×C₂₇×C₈₁, C₃×C₅×C₂₇², …
maximal quotient of	C₂×C₅×C₂₇², C₅×C₂₇×C₈₁, C₃×C₅×C₂₇², …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3645, 2);

C5×C272 #16813951