#34203371

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1379 ⋊ C₂
Sylow	C₇₉ ⋊ #12614

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1379	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1379	D₇₉
hypocenter	C₇₉	#12614
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 115 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)(84,86,88,97,89,98,101,111,90,99,102,112,87,96,100,110)(85,92,94,106,93,105,108,114,95,107,109,115,91,103,104,113);(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)(80,81,82,83)(85,91,95,93)(92,103,107,105)(94,104,109,108)(106,113,115,114);(84,85,88,94,89,93,101,108,90,95,102,109,87,91,100,104)(86,92,97,106,98,105,111,114,99,107,112,115,96,103,110,113)

Polycyclic presentation:

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

Conjugacy classes & character table

2296 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84670);