#27532876

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#458 ⋊ C₂
Sylow	C₅₃ ⋊ #32901

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×#458	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#16811458	C₂
hypercenter	C₂×#458	D₅₃
hypocenter	C₅₃	#32901
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#458

3706 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 251 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 73 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)(55,57)(58,62,63,59)(60,67,68,64)(61,69,70,65)(66,72,73,71);(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)(58,61)(59,65)(60,66)(62,69)(63,70)(64,71)(67,72)(68,73);(58,60)(59,64)(61,66)(62,67)(63,68)(65,71)(69,72)(70,73);(54,55)(56,57)(58,59,63,62)(60,64,68,67)(61,65,70,69)(66,72,73,71)

Polycyclic presentation:

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

Conjugacy classes & character table

881 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 318352);