#27289522

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10763

Subgroups

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	#10763
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×Q₈

1258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(62,64,66,72)(63,68,67,70)(65,74,73,71)(69,77,75,76);(54,55,56,57)(59,61)(63,67)(65,73)(68,70)(71,74);(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)(62,63,65,69,66,67,73,75)(64,70,71,77,72,68,74,76)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,i^a=i^52,f^b=fh,f^c=fh

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74998);