#34179316

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #8248

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1500	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	#121×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1500	D₇₉
hypocenter	C₇₉	#8248
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1500

392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 5056

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

Polycyclic presentation:

2^8,79|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,b^a=bd,d^a=dg,f^a=fgh,i^a=i^78,c^b=cg,d^b=dh,i^c=i^78

Conjugacy classes & character table

1534 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 60615);