#27537676

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33486

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#33486
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄

662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 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)(55,59)(58,60);(56,61)(58,60);(54,56)(55,58)(57,61)(59,60);(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)(56,60)(57,59)(58,61)(62,63,64,67,65,68,70,73,66,69,71,74,72,75,76,77)

Polycyclic presentation:

2^8,53|a^2=f,f^2=g,g^2=h,b^a=be,d^a=de,i^a=i^52,c^b=ce

Conjugacy classes & character table

2216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323152);