#27274071

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8010

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#2920	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#8010
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #2920

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 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)(54,57,58,66)(55,60,61,69)(56,63,64,65)(59,67,62,68);(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)(54,56)(55,59)(57,65)(58,64)(60,68)(61,62)(63,66)(67,69)(70,72,74,81,75,82,73,80)(71,76,77,79,78,85,83,84);(54,55)(56,62,64,59)(57,60)(58,61)(63,68,65,67)(66,69)(70,71,73,77,75,78,74,83)(72,79,81,76,82,84,80,85)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh,i^b=i^52

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 59547);