#34180812

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1512 ⋊ C₂
Sylow	C₇₉ ⋊ #8528

Subgroups

	subgroup	G/H
center	C₂²	#19098418
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1512	C₂
socle	C₂²×C₇₉	#379
prime radical	#23777752	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1512	D₇₉
hypocenter	C₇₉	#8528
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1512

728 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)(80,81)(82,85)(83,87)(84,86)(88,91)(89,95)(90,99)(92,106)(93,104)(94,108)(96,112)(97,102)(98,114)(100,117)(101,116)(103,107)(105,115)(109,118)(110,113)(111,119);(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)(82,87)(83,85)(84,86)(88,90)(89,94)(91,99)(92,105)(93,101)(95,108)(96,111)(97,109)(98,113)(100,107)(102,118)(103,117)(104,116)(106,115)(110,114)(112,119);(81,85)(82,83)(86,87)(88,89)(90,98,101,111)(91,102,104,95)(92,96)(93,97)(94,100,109,105)(99,114)(103,112,106,110)(107,113)(108,115)(116,119)(117,118)

Polycyclic presentation:

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

Conjugacy classes & character table

1054 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 62111);