Dic₆×C₅₃ #11929671

order	1272
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₆×C₅₃
unambiguous semidirect product	C₃⋊(Q₈×C₅₃)

Splittings

Hall	C₃×C₅₃ ⋊ Q₈
Sylow	C₃ ⋊ Q₈×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, prime radical	C₂×C₃×C₅₃	C₂²
hypercenter	C₄×C₅₃	D₃
hypocenter	C₃	Q₈×C₅₃
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅₃ ◁ C₄×C₅₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1272

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

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

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, 20);

Dic6×C53 #11929671