#34189062

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#2562 ⋊ C₂
Sylow	C₇₉ ⋊ #10005

Subgroups

	subgroup	G/H
center	C₂²	#19098471
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2562	C₂
socle	C₂²×C₇₉	#392
prime radical	#23773491	C₂
hypercenter	#2562	D₇₉
hypocenter	C₇₉	#10005
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2562

566 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 103
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 103 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,82,85,81)(83,87,84,86)(88,90,92,98)(89,94,96,103)(91,100,99,97)(93,101,95,102);(80,81)(82,85)(83,87)(84,86)(89,93)(91,99)(94,102)(95,96)(97,100)(101,103);(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(81,86)(82,87)(83,84)(88,89)(90,94)(91,93)(92,96)(95,99)(97,102)(98,103)(100,101)

Polycyclic presentation:

2^8,79|c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,i^a=i^78,c^b=cf,d^b=dg,d^c=dh,e^c=egh

Conjugacy classes & character table

1516 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 70361);