#27539209

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3176 ⋊ C₂
Sylow	C₅₃ ⋊ #33679

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3176	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#3176	D₅₃
hypocenter	C₅₃	#33679
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324685);