#34204108

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#2120 ⋊ C₂
Sylow	C₇₉ ⋊ #12766

Subgroups

	subgroup	G/H
center	C₄	#19098751
commutator	C₂×D₄×C₇₉	C₂²×C₄
Frattini	C₂×#41	C₂²×D₇₉
Fitting	C₇₉×#2120	C₂
socle	C₂×C₇₉	#1790
prime radical	#23777538	C₂
cosocle	C₂×#41×C₇₉	C₂³
hypercenter	#2120	D₇₉
hypocenter	C₇₉	#12766
LCS	G ▷ C₂×D₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2120

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 points:

(2,9)(3,14)(8,25)(10,11)(12,30)(13,27)(15,19)(16,22)(17,24)(18,29)(26,28)(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)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110);(1,3)(2,8)(4,19)(5,22)(6,24)(7,14)(9,26)(10,31)(11,27)(12,28)(13,18)(15,21)(16,23)(17,20)(25,30)(29,32)(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)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111);(1,2,6,11,7,12,20,29)(3,13,17,26,14,31,24,25)(4,18,5,9,21,10,23,30)(8,22,27,19,28,16,32,15)

Polycyclic presentation:

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

Conjugacy classes & character table

1048 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 85407);