#27293968

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×Q₈⋊C₁₆ ⋊ C₂
Sylow	C₅₃ ⋊ #11622

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#13133192	C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	Q₈⋊C₁₆	D₅₃
hypocenter	C₅₃	#11622
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

320 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 6784

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

Polycyclic presentation:

2^8,53|a^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=cfh,d^b=df,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79444);