#27399456

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1784 ⋊ C₂
Sylow	C₅₃ ⋊ D₄×#101

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1784	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×C₂²⋊D₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1784	D₅₃
hypocenter	C₅₃	D₄×#101
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1784

1378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

887 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 184932);