#27295836

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12044

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×Q₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄×D₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×Q₁₆	D₅₃
hypocenter	C₅₃	#12044
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×Q₁₆

222 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 13568

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

Polycyclic presentation:

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

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81312);