#27304620

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1397 ⋊ C₂
Sylow	C₅₃ ⋊ #13640

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×#1397	C₂
prime radical	C₂²×D₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1397	D₅₃
hypocenter	C₅₃	#13640
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

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

normal subgroups only

This group has 194 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,57,60)(55,59,58,61)(63,68)(69,78)(70,79)(71,80)(81,88)(82,89)(83,90)(91,93);(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)(55,58)(59,61);(54,55)(56,59)(57,58)(60,61)(62,63,64,69,65,70,72,81,66,71,73,82,75,83,84,91,67,68,74,78,76,79,85,88,77,80,86,89,87,90,92,93)

Polycyclic presentation:

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

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90096);