#27355845

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

Aliases

No aliases for this group.

Splittings

Fitting	C₈×D₈×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #18066

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285291	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×D₈	D₅₃
hypocenter	C₅₃	#18066
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₈

722 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)(62,65)(63,69)(64,71)(66,74)(67,73)(68,75)(70,76)(72,77);(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)(62,64,66,72,67,63,68,70)(65,69,74,76,73,71,75,77);(55,58)(56,61)(59,60)(65,73)(69,71)(74,75)(76,77);(54,55)(56,58)(57,60)(59,61)(62,63,66,70,67,64,68,72)(65,69,74,76,73,71,75,77)

Polycyclic presentation:

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

Conjugacy classes & character table

1526 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 141321);