#34180863

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1468 ⋊ C₂
Sylow	C₇₉ ⋊ #8539

Subgroups

	subgroup	G/H
center	C₂²	#19098418
commutator	C₂⁴×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1468	C₂
socle	C₂²×C₇₉	#379
prime radical	#23777038	C₂
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1468	D₇₉
hypocenter	C₇₉	#8539
LCS	G ▷ C₂⁴×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1468

1008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1288 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 62162);