#27489519

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2616 ⋊ C₂
Sylow	C₅₃ ⋊ #29604

Subgroups

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2616	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂×#16811187	C₂
hypercenter	#2616	D₅₃
hypocenter	C₅₃	#29604
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ #2616

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 81 points:

(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)(60,64)(63,65)(66,68)(67,73)(69,75)(70,76)(71,81)(72,78)(74,80)(77,79);(55,57)(58,60)(59,63)(61,64)(62,65)(66,68,69,75)(67,72,71,80)(70,79,77,76)(73,74,81,78);(54,55)(56,57)(59,62)(63,65)(67,71)(68,75)(74,78)(76,79);(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)(58,59)(60,63)(61,62)(64,65)(66,67)(68,73)(69,71)(70,78)(72,76)(74,77)(75,81)(79,80)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 274995);