#27315297

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #14692

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₅₃×#384	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#384	D₅₃
hypocenter	C₅₃	#14692
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

2286 conjugacy classes of subgroups; only the 947 normal subgroups are shown.

normal subgroups only

This group has 2286 conjugacy classes of subgroups — too many to draw. Only its 947 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) = 69
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 69 points:

(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)(66,67,68,69);(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)(62,63,64,65);(58,59,60,61);(54,55,56,57)(59,61)(63,65)(67,69)

Polycyclic presentation:

2^8,53|a^2=h,b^2=e,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dg,i^d=i^52

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 100773);