#27271987

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ C₄⋊C₆₄

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₆₄
Fitting	C₂×C₅₃×C₆₄	C₂
socle, prime radical	C₂²×C₅₃	C₂×C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	C₄⋊C₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

74 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:

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

Polycyclic presentation:

2^8,53|a^2=d,b^2=c,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bc,i^b=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, 57463);