#27472302

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1804 ⋊ C₂
Sylow	C₅₃ ⋊ #28165

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1804	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	#1804	D₅₃
hypocenter	C₅₃	#28165
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1804

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 81 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)(58,60,61,65)(59,63,62,64)(68,75)(71,79)(76,81)(78,80);(55,57)(67,72)(69,77)(71,78)(73,74)(76,81)(79,80);(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)(54,55,56,57)(59,62)(63,64)(66,68)(67,71)(69,76)(70,75)(72,78)(73,80)(74,79)(77,81);(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)(58,59)(60,64)(61,62)(63,65)(66,67)(68,71)(69,72)(70,74)(73,77)(75,79)(76,78)(80,81)

Polycyclic presentation:

2^8,53|b^2=g,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,i^a=i^52,c^b=cg,d^b=dh,i^b=i^52,f^c=fh

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 257778);