#27293587

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11536

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1324	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	#1324	D₅₃
hypocenter	C₅₃	#11536
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1324

376 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 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)(54,56,58,57)(55,59,60,61)(62,65,67,78)(63,69,71,86)(64,73,75,76)(66,77,81,93)(68,82,83,84)(70,85,79,92)(72,88,87,90)(74,89,80,91);(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)(62,64,67,75)(63,68,71,83)(65,76,78,73)(66,80,81,74)(69,84,86,82)(70,87,79,72)(77,89,93,91)(85,88,92,90);(54,55,57,59,58,60,56,61)(62,63)(64,72,75,87)(65,69)(66,79,81,70)(67,71)(68,74)(73,88,76,90)(77,92,93,85)(78,86)(80,83)(82,89)(84,91)

Polycyclic presentation:

2^8,53|a^2=e,b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=efh,i^b=i^52,i^c=i^52,e^d=eh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79063);