#27271971

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×C₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ #7537

Subgroups

	subgroup	G/H
center	C₂²	D₁₆₉₆
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle	C₂²×C₅₃	D₃₂
prime radical	C₂×D₁₆₉₆	C₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	#7537
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 2 generators on 121 points:

(1,2)(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)(59,65)(60,71)(61,72)(62,77)(63,81)(66,67)(68,93)(69,95)(70,87)(73,103)(74,106)(75,99)(76,84)(78,112)(79,102)(80,85)(82,110)(83,86)(88,108)(89,107)(90,92)(91,114)(94,118)(96,97)(98,104)(100,101)(105,121)(109,116)(111,115)(113,117)(119,120);(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)(58,59)(60,65)(61,76)(62,80)(63,83)(64,70)(66,71)(67,72)(68,77)(69,81)(73,93)(74,95)(75,87)(78,111)(79,109)(82,113)(84,97)(85,96)(86,104)(88,103)(89,106)(90,99)(91,112)(92,102)(94,110)(98,114)(100,108)(101,107)(105,118)(115,121)(116,120)(117,119)

Polycyclic presentation:

2^8,53|a^2=d,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^a=i^52,c^b=ce,e^b=ef,f^b=fg,g^b=gh,i^b=i^52

Conjugacy classes & character table

3398 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57447);