#34180127

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1500 ⋊ C₂
Sylow	C₇₉ ⋊ #8413

Subgroups

	subgroup	G/H
center	C₂²	#19098418
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1500	C₂
socle	C₂²×C₇₉	#379
prime radical	#23777415	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1500	D₇₉
hypocenter	C₇₉	#8413
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1500

702 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)(88,91,93,104)(89,95,96,109)(90,99,101,116)(92,105,106,103)(94,107,98,112)(97,111,110,102)(100,117,118,115)(108,114,113,119);(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)(80,81)(82,86)(83,85)(84,87)(88,90)(89,94)(91,99)(92,100)(93,101)(95,107)(96,98)(97,108)(102,119)(103,115)(104,116)(105,117)(106,118)(109,112)(110,113)(111,114);(81,84,85,86)(88,89)(90,98,101,94)(91,102,104,111)(92,97)(93,96)(95,103,109,105)(99,114)(100,113,118,108)(106,110)(107,115)(112,117)(116,119)

Polycyclic presentation:

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

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 61426);