#27305960

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#386 ⋊ C₂
Sylow	C₅₃ ⋊ #13867

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₅₃×#386	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₄×D₅₃	C₂
hypercenter	C₂×#386	D₅₃
hypocenter	C₅₃	#13867
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#386

7150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 69 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)(59,61);(63,65)(67,69);(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)(59,61)(66,67)(68,69);(54,55,56,57)(58,59)(60,61)(62,63)(64,65)

Polycyclic presentation:

2^8,53|a^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,i^b=i^52,i^d=i^52

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 91436);