#27411510

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23470

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#2310	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#2310	D₅₃
hypocenter	C₅₃	#23470
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2310

1178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 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)(62,66,67,78)(63,70,71,86)(64,73,74,90)(65,75,76,77)(68,80,81,93)(69,83,84,85)(72,87,88,89)(79,91,82,92);(62,65,67,76)(63,69,71,84)(64,72,74,88)(66,77,78,75)(68,79,81,82)(70,85,86,83)(73,89,90,87)(80,92,93,91);(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)(54,56,57,61)(55,58,59,60)(62,64)(63,68)(65,72)(66,73)(67,74)(69,82)(70,80)(71,81)(75,87)(76,88)(77,89)(78,90)(79,84)(83,92)(85,91)(86,93);(54,55,57,59)(56,60,61,58)(62,63)(64,68)(65,69)(66,70)(67,71)(72,82)(73,80)(74,81)(75,83)(76,84)(77,85)(78,86)(79,88)(87,92)(89,91)(90,93)

Polycyclic presentation:

2^8,53|a^2=e,b^2=e,c^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,i^b=i^52,d^c=dh

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196986);