#27304615

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13639

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Fitting	C₅₃×C₂²⋊C₃₂	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂²⋊C₃₂	D₅₃
hypocenter	C₅₃	#13639
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²⋊C₃₂

202 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 101 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90091);