#27277146

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8644

Subgroups

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#8644
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

2138 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62622);