#27398015

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1979 ⋊ C₂
Sylow	C₅₃ ⋊ #22285

Subgroups

	subgroup	G/H
center	C₂	#20277439
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1979	C₂
socle	C₂×C₅₃	#2328
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1979	D₅₃
hypocenter	C₅₃	#22285
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1979

772 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)(54,58,79,107,60,81,59,80)(55,63,65,89,66,90,64,78)(56,69,71,99,72,100,70,98)(57,75,102,117,77,105,76,104)(61,83,91,112,84,97,67,92)(62,86,73,101,88,111,87,106)(68,94,95,115,96,116,74,103)(82,108,110,114,93,113,85,109);(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)(54,57,60,77)(55,62)(56,68)(58,75,81,105)(59,76,79,102)(61,82,84,93)(63,86)(64,87)(65,73)(66,88)(67,85,91,110)(69,94)(70,74)(71,95)(72,96)(78,106)(80,104,107,117)(83,108,97,113)(89,101)(90,111)(92,109,112,114)(98,103)(99,115)(100,116);(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)(54,56,59,70,60,72,79,71)(55,61,64,67,66,84,65,91)(57,74,102,68,77,95,76,96)(58,69,80,98,81,100,107,99)(62,85,73,82,88,110,87,93)(63,83,78,92,90,97,89,112)(75,103,117,94,105,115,104,116)(86,109,101,108,111,114,106,113);(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)(56,67)(57,73,77,87)(58,78)(59,65)(60,66)(61,70)(62,76,88,102)(63,80)(64,79)(68,93,96,82)(69,97)(71,84)(72,91)(74,85,95,110)(75,86,105,111)(81,89)(83,100)(90,107)(92,98)(94,114,116,109)(99,112)(101,117,106,104)(103,113,115,108)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 183491);