#27385352

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1874 ⋊ C₂
Sylow	C₅₃ ⋊ #20914

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1874	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂×#16811189	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1874	D₅₃
hypocenter	C₅₃	#20914
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1874

1970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

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

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=egh,e^2=h,g^2=h,b^a=be,c^a=cf,e^b=eh,g^b=gh,d^c=dh,i^d=i^52

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 170828);