#27539453

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2242 ⋊ C₂
Sylow	C₅₃ ⋊ #33706

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2242	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₈×D₅₃	C₂²
hypercenter	#2242	D₅₃
hypocenter	C₅₃	#33706
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2242

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(64,78)(65,81)(70,89)(71,76)(79,83)(80,82)(90,93)(91,92);(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)(55,59)(58,61)(62,65,66,82,85,79,67,78,69,81,86,80,68,83,77,64)(63,71,72,91,88,93,73,70,75,76,87,92,74,90,84,89);(54,56,57,60)(55,58,59,61)(62,64,77,83,68,80,86,81,69,78,67,79,85,82,66,65)(63,70,84,93,74,91,87,71,75,89,73,90,88,92,72,76);(54,55)(56,58)(57,59)(60,61)(62,63)(64,76)(65,70)(66,84)(67,87)(68,88)(69,75)(71,78)(72,77)(73,86)(74,85)(79,91)(80,90)(81,89)(82,93)(83,92)

Polycyclic presentation:

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

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324929);