#27400240

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1740 ⋊ C₂
Sylow	C₅₃ ⋊ #22478

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1740	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×C₂²⋊D₅₃	C₂³
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	#1740	D₅₃
hypocenter	C₅₃	#22478
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1740

1132 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)(55,58)(59,61)(62,66,67,64)(63,70,71,68)(65,74,75,72)(69,77,73,76);(54,56,57,60)(55,59,58,61)(62,65)(63,69)(64,72)(66,74)(67,75)(68,76)(70,77)(71,73);(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)(56,59)(57,58)(60,61)(62,64,67,66)(63,68,71,70)(65,72,75,74)(69,76,73,77);(55,58)(59,61)(62,63)(64,68)(65,73,75,69)(66,70)(67,71)(72,77,74,76)

Polycyclic presentation:

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

Conjugacy classes & character table

1199 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 185716);