#27290900

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11024

Subgroups

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#103×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#103	D₅₃
hypocenter	C₅₃	#11024
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#103

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

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

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76376);