#27540855

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2179 ⋊ C₂
Sylow	C₅₃ ⋊ #33855

Subgroups

	subgroup	G/H
center	C₂	D₈×D₂₁₂
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2179	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2179	D₅₃
hypocenter	C₅₃	#33855
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2179

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 117 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)(54,57)(55,63)(56,70)(58,80)(59,76)(60,84)(61,77)(62,88)(64,94)(65,74)(66,97)(67,92)(68,100)(69,79)(71,105)(72,103)(73,110)(75,82)(78,91)(81,108)(83,115)(85,109)(86,104)(87,93)(89,99)(90,102)(95,113)(96,117)(98,116)(101,112)(106,114)(107,111);(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)(54,57)(55,63)(56,69)(58,75)(59,84)(60,76)(61,77)(62,87)(64,91)(65,97)(66,74)(67,92)(68,99)(70,79)(71,103)(72,105)(73,106)(78,94)(80,82)(81,115)(83,108)(85,104)(86,109)(88,93)(89,100)(90,107)(95,117)(96,113)(98,112)(101,116)(102,111)(110,114);(54,56,61,73)(55,62,67,90)(57,69,77,106)(58,79,83,114)(59,72,60,85)(63,87,92,107)(64,93,96,111)(65,89,66,98)(68,95,101,78)(70,108,110,75)(71,81,86,82)(74,112,97,100)(76,104,84,105)(80,103,115,109)(88,113,102,91)(94,99,117,116);(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)(54,55,61,67)(56,68,73,101)(57,74,77,97)(58,78,83,95)(59,66,60,65)(62,86,90,71)(63,84,92,76)(64,81,96,82)(69,102,106,88)(70,107,110,87)(72,111,85,93)(75,113,108,91)(79,98,114,89)(80,94,115,117)(99,105,116,104)(100,103,112,109)

Polycyclic presentation:

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

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326331);