#27288600

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10585

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1295	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#1295	D₅₃
hypocenter	C₅₃	#10585
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1295

382 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 77 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)(54,57,58,56)(55,59,60,61)(63,66)(64,69)(67,68)(71,74)(72,77)(75,76);(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)(54,56,58,57)(55,59,60,61)(64,69)(66,67)(70,71)(72,74)(73,75)(76,77);(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)(54,55,58,60)(56,61,57,59)(62,63,65,68)(64,66,69,67)(72,77)(74,76)

Polycyclic presentation:

2^8,53|a^2=dg,b^2=d,c^2=d,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,i^a=i^52,c^b=cf,e^b=eh,f^b=fgh,i^b=i^52,e^c=eg,f^c=fgh,i^c=i^52

Conjugacy classes & character table

1022 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74076);