#27276753

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8558

Subgroups

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#8558
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

682 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)(66,68,69,73)(67,70,71,72);(54,55,56,57);(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)(55,57)(58,59,60,62,61,63,64,65)(66,67)(68,72,73,70)(69,71)

Polycyclic presentation:

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

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62229);