#27437805

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3511 ⋊ C₂
Sylow	C₅₃ ⋊ #25618

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3511	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3511	D₅₃
hypocenter	C₅₃	#25618
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3511

1114 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)(54,56,57,61)(55,59,58,60)(62,66)(63,64)(65,70)(67,74)(68,76)(69,72)(71,73)(75,77);(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)(62,65,67,64,68,72,75,73)(63,66,69,74,71,76,70,77);(55,58)(59,60)(62,64)(63,66)(65,75)(67,72)(68,73)(69,74)(70,77)(71,76);(54,55)(56,60)(57,58)(59,61)(62,63)(64,66)(65,74)(67,70)(68,71)(69,75)(72,77)(73,76)

Polycyclic presentation:

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

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 223281);