#27295840

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1331 ⋊ C₂
Sylow	C₅₃ ⋊ #12045

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1331	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×#13133158	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1331	D₅₃
hypocenter	C₅₃	#12045
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1331

254 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

(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)(54,56,57,61,58,55,59,60)(63,68)(67,75)(69,76)(70,74);(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)(63,67)(64,71)(66,72)(68,75)(69,70)(73,77)(74,76);(54,55,57,60,58,56,59,61)(62,63,65,68)(64,67,73,75)(66,74,72,70)(69,77,76,71)

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81316);