#27271897

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7510

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₄×C₃₂
Fitting	C₄×C₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#7510
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₄×C₃₂

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 generators on 121 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)(54,55,56,57)(59,65)(66,80)(67,81)(68,82)(69,83)(84,100)(85,101)(86,102)(87,103)(88,104)(89,105)(106,115)(107,116)(108,117)(109,118)(119,121);(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)(58,59,60,66,61,67,70,84,62,68,71,85,74,87,90,106,63,69,72,86,75,88,91,107,77,89,93,108,96,109,110,119,64,65,73,80,76,81,92,100,78,82,94,101,97,103,111,115,79,83,95,102,98,104,112,116,99,105,113,117,114,118,120,121)

Polycyclic presentation:

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

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57373);