#34182175

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #8842

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579131
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#104×C₇₉	C₂
socle	C₂²×C₇₉	#385
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#104	D₇₉
hypocenter	C₇₉	#8842
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#104

420 conjugacy classes of subgroups.

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)(81,85)(82,86)(88,91,94,90)(89,96,99,95)(92,103,93,101)(97,102,98,100);(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)(80,82,84,85,83,86,87,81)(88,90,94,91)(89,95,99,96)(92,103,93,101)(97,102,98,100);(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)(80,81,84,82,83,85,87,86)(88,89,94,99)(90,100,91,102)(92,98,93,97)(95,103,96,101)

Polycyclic presentation:

2^8,79|a^2=f,b^2=f,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,i^a=i^78,c^b=ch,d^b=dg,i^b=i^78,e^c=eg

Conjugacy classes & character table

2236 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 63474);