#27299242

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#100×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #12622

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133131
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₄×#100×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#13133185	C₂²
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	C₄×#100	D₅₃
hypocenter	C₅₃	#12622
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ C₄×#100

636 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 73
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 73 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)(54,56,57,55)(59,64)(63,71)(65,72)(66,73);(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)(58,60)(59,63)(61,69,70,67)(62,68)(64,71)(65,73,72,66);(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)(54,55,57,56)(58,59,62,64)(60,66,68,73)(61,65,70,72)(63,69,71,67)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84718);