#27289136

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10672

Subgroups

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1957	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#127×C₅₃	C₂²
hypercenter	#1957	D₅₃
hypocenter	C₅₃	#10672
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 77 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)(63,68)(64,65)(67,69)(71,74)(72,77)(75,76);(54,56,57,61)(55,58,59,60)(63,67)(64,65)(68,69)(70,71)(72,74)(73,76)(75,77);(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)(54,55,57,59)(56,60,61,58)(62,63,65,67,66,69,64,68)

Polycyclic presentation:

2^8,53|a^2=dg,b^2=de,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,i^a=i^52,c^b=cf,d^b=dg,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fh

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74612);