#34185043

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1805 ⋊ C₂
Sylow	C₇₉ ⋊ #9306

Subgroups

	subgroup	G/H
center	C₂²	#19098440
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1805	C₂
socle	C₂²×C₇₉	#386
prime radical	#19099113	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1805	D₇₉
hypocenter	C₇₉	#9306
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1805

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 119 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)(80,82,84,83)(81,86,87,85)(88,91,93,103)(89,95,96,110)(90,98,99,113)(92,105,106,102)(94,107,109,118)(97,111,104,101)(100,117,115,114)(108,119,116,112);(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)(80,81,82,86,84,87,83,85)(88,90,93,99)(89,94)(91,98,103,113)(92,100,106,115)(95,107)(96,109)(97,108)(101,112)(102,114,105,117)(104,116)(110,118)(111,119);(81,85)(82,83)(86,87)(88,89)(90,94,99,109)(91,101)(92,104)(93,96)(95,102)(97,106)(98,112,113,119)(100,116,115,108)(103,111)(105,110)(107,117,118,114)

Polycyclic presentation:

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

Conjugacy classes & character table

1531 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 66342);