#27289883

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10834

Subgroups

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#336	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#336	D₅₃
hypocenter	C₅₃	#10834
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 73 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 75359);