#27276162

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×Q₈⋊C₈ ⋊ C₂
Sylow	C₅₃ ⋊ #8444

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×Q₈⋊C₈	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂×#16811295	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#8444
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

1116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 73 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)(66,68,71,72)(67,69,73,70);(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)(59,63)(60,61)(64,65)(66,67)(68,69)(70,72)(71,73);(54,55,56,57)(58,59,61,63,62,65,60,64)(67,72)(68,73)(69,70)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,i^b=i^52,d^c=dg,e^c=eg

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61638);