D₅₃×#2056 #27412031

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2056

Splittings

Fitting	C₅₃×#2056 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2056

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2056	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
hypercenter	#2056	D₅₃
hypocenter	C₅₃	C₂×#2056
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2056

1486 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:

(2,4)(6,8)(9,12,13,11,14,19,22,20)(10,16,17,21,18,24,23,15)(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);(5,6)(7,8)(9,11,14,20)(10,15,18,21)(12,22,19,13)(16,17,24,23);(1,2)(3,4)(9,10,14,18)(11,16,20,24)(12,21,19,15)(13,23,22,17);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 197507);

D53×#2056 #27412031