#27293145

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11415

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2163	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	#97
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#11415
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #2163

246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 85 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)(56,62)(59,65)(63,68)(66,69);(54,56)(55,59)(57,63)(58,62)(60,66)(61,65)(64,68)(67,69)(70,71,72,77,73,75,79,82,74,76,78,83,80,81,85,84);(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)(54,55,57,60,58,61,64,67)(56,59,63,66,62,65,68,69)(71,75,76,81)(72,78)(77,84,83,82)(79,85)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,b^a=bg,e^a=eh,i^a=i^52,c^b=cf,d^b=dh

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, 78621);