#34184166

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1545 ⋊ C₂
Sylow	C₇₉ ⋊ #9174

Subgroups

	subgroup	G/H
center	C₂²	#19098439
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1545	C₂
socle	C₂²×C₇₉	#386
prime radical	#23772876	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1545	D₇₉
hypocenter	C₇₉	#9174
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1545

516 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:

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

Polycyclic presentation:

2^8,79|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fg,i^c=i^78

Conjugacy classes & character table

1291 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 65465);