#27289085

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10663

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1985	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	C₂²⋊Q₈
hypercenter	#1985	D₅₃
hypocenter	C₅₃	#10663
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

320 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 99 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:

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

Polycyclic presentation:

2^8,53|a^2=dg,b^2=de,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fgh,i^a=i^52,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74561);