#27550411

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #34552

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₄×#117×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₂⁴
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#34552
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

1962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(62,64,65,68)(63,67,66,69);(54,56,57,60)(55,59,58,61);(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)(55,58)(59,61)(70,72,73,77)(71,74,75,76);(63,66)(67,69)(70,71,73,75)(72,76,77,74);(54,55)(56,59)(57,58)(60,61)(62,63)(64,67)(65,66)(68,69)

Polycyclic presentation:

2^8,53|b^2=h,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,i^c=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, 335887);