#27311969

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1436 ⋊ C₂
Sylow	C₅₃ ⋊ #14398

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1436	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁴×D₅₃	C₂³
hypercenter	#1436	D₅₃
hypocenter	C₅₃	#14398
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ #1436

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 77 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)(64,69)(66,68)(70,73,74,72)(71,75,76,77);(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)(59,61)(63,67)(64,69)(71,76)(75,77);(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)(58,59)(60,61)(62,64)(63,66)(65,69)(67,68)(70,72,74,73)(71,75,76,77);(54,55,56,57)(62,63)(64,68)(65,67)(66,69)(70,71)(72,75)(73,77)(74,76)

Polycyclic presentation:

2^8,53|a^2=h,b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 97445);