#27353975

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1582 ⋊ C₂
Sylow	C₅₃ ⋊ #17902

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1582	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×D₄×D₅₃	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1582	D₅₃
hypocenter	C₅₃	#17902
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1582

706 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:

(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)(63,68)(67,73)(69,75)(74,77);(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)(62,64,65,70,66,71,72,76)(63,67,69,74,68,73,75,77);(55,58)(56,61)(59,60);(54,55,57,60)(56,58,61,59)(62,63,65,69,66,68,72,75)(64,67,70,74,71,73,76,77)

Polycyclic presentation:

2^8,53|a^2=f,c^2=fg,e^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,e^b=eg,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 139451);