#34196588

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #11277

Subgroups

	subgroup	G/H
center	C₄	#19098446
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×H(3,ℤ/4)×C₇₉	C₂
socle	C₂×C₇₉	#1442
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₇₉
hypocenter	C₇₉	#11277
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

592 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:

(1,2)(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)(80,83)(81,87,88,94)(82,91)(84,98)(85,96)(86,100,90,106)(89,105,104,103)(92,109)(93,108)(95,110)(97,99)(101,111,107,102);(80,82)(81,86)(83,95,96,109)(84,92)(85,93)(87,102,94,111)(88,90)(89,101)(91,97,108,98)(99,110)(100,103,106,105)(104,107);(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)(82,90,93,86)(83,94)(84,89)(85,88)(87,96)(91,100,108,106)(92,107,110,101)(95,102,109,111)(97,105)(98,103)(99,104)

Polycyclic presentation:

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

Conjugacy classes & character table

1762 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 77887);