#34181491

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #8678

Subgroups

	subgroup	G/H
center	C₂²	#19098421
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1517	C₂
socle	C₂²×C₇₉	#380
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1517	D₇₉
hypocenter	C₇₉	#8678
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1517

418 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)(81,85)(83,87)(84,86)(88,91)(89,95)(90,100)(92,104)(93,105)(94,107)(96,109)(97,103)(98,110)(99,113)(101,115)(102,116)(106,118)(108,117)(111,114)(112,119);(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)(81,84)(85,86)(88,90,99,112,93,102,92,101)(89,94,106,111,97,108,96,98)(91,100,113,119,105,116,104,115)(95,107,118,114,103,117,109,110);(80,81)(82,84)(83,85)(86,87)(88,89)(90,98)(91,103,105,95)(92,106)(93,97)(94,101)(96,99)(100,114,116,110)(102,111)(104,109,113,118)(107,115,117,119)(108,112)

Polycyclic presentation:

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

Conjugacy classes & character table

1294 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 62790);