#34203597

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12657

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	C₂×#97
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#12657
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #2150

334 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)(84,87,88,89,90,102,103,105)(85,92,93,104,95,109,110,94)(86,97,112,113,99,100,98,101)(91,106,111,114,107,115,96,108);(81,83)(84,86,89,100,88,98,102,113,90,99,105,97,103,112,87,101)(85,91,94,106,93,96,92,108,95,107,104,115,110,111,109,114);(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)(84,85)(86,96,99,111)(87,92)(88,93)(89,104)(90,95)(91,98,107,112)(94,105)(97,108,100,114)(101,115,113,106)(102,109)(103,110)

Polycyclic presentation:

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

Conjugacy classes & character table

2248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84896);