#27350573

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1487 ⋊ C₂
Sylow	C₅₃ ⋊ #17454

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1487	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×#13133265	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1487	D₅₃
hypocenter	C₅₃	#17454
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1487

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,e^a=eg,e^b=eg,d^c=dg,i^d=i^52

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 136049);