D₁₂×C₅₃ #11929673

order	1272
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅₃ · Schur multiplier (H₂) = C₂

metacyclic
supersolvable
metabelian
solvable
monomial

Aliases

direct product	D₁₂×C₅₃
unambiguous semidirect product	C₄⋊(D₃×C₅₃)

Splittings

Fitting	C₃×C₄×C₅₃ ⋊ C₂
Hall	C₃×C₅₃ ⋊ D₄
Sylow	C₃ ⋊ D₄×C₅₃

Subgroups

	subgroup	G/H
center	C₂×C₅₃	C₂×D₃
commutator, cosocle	C₂×C₃	C₂²×C₅₃
Frattini	C₂	C₂×D₃×C₅₃
Fitting	C₃×C₄×C₅₃	C₂
socle	C₂×C₃×C₅₃	C₂²
hypercenter	C₄×C₅₃	D₃
hypocenter	C₃	D₄×C₅₃
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅₃ ◁ C₄×C₅₃

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 636

A minimal faithful permutation representation — 2 generators on 60 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1272, 22);

D12×C53 #11929673