#34193955

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #10811

Subgroups

	subgroup	G/H
center	C₄	C₄×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×H(3,ℤ/4)×C₇₉	C₂
socle	C₂×C₇₉	#1434
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₇₉
hypocenter	C₇₉	#10811
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

644 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)(80,83,84,98,86,100,85,99)(81,88,89,97,91,104,90,105)(82,93,94,108,95,110,96,109)(87,101,92,106,103,107,102,111);(80,82,85,96,86,95,84,94)(81,87,89,92,91,103,90,102)(83,93,99,109,100,110,98,108)(88,101,97,106,104,107,105,111);(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)(82,92,95,102)(83,97)(84,89)(85,90)(86,91)(87,94,103,96)(88,99)(93,107,110,101)(98,104)(100,105)(106,108,111,109)

Polycyclic presentation:

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

Conjugacy classes & character table

1774 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 75254);