#27399484

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	D₄×M₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #22412

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719921
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	D₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×#16811294	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	D₄×M₁₆	D₅₃
hypocenter	C₅₃	#22412
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×M₁₆

1462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

1355 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 184960);