#27272014

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7555

Subgroups

	subgroup	G/H
center, Frattini	C₂⁵	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Fitting	C₂³×C₂²⋊C₄×C₅₃	C₂
socle, cosocle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁶×C₅₃	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#7555
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

4566 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)(67,69)(71,73);(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)(58,59,60,61)(63,65)(70,71)(72,73);(54,55,56,57)(62,63)(64,65)(66,67)(68,69)

Polycyclic presentation:

2^8,53|a^2=g,b^2=h,b^a=bd,c^a=ce,c^b=cf,i^b=i^52

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57490);