D₅₃×#2227 #27538717

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

D₅₃×#2227

Splittings

Fitting	C₅₃×#2227 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2227

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2227	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2227	D₅₃
hypocenter	C₅₃	C₂×#2227
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 77 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324193);

D53×#2227 #27538717