#34195090

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#421×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #11026

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579133
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#421×C₇₉	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂²×#14579181	C₂
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#421	D₇₉
hypocenter	C₇₉	#11026
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#421

1278 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)(85,91)(86,95)(88,103)(89,101)(90,105)(92,98)(93,111)(94,112)(96,114)(97,113)(99,102)(100,104)(106,115)(107,110)(108,109);(81,83)(84,86)(85,90)(87,99)(88,102)(89,97)(91,107)(92,94)(93,110)(95,100)(96,104)(98,115)(101,114)(103,113)(105,108)(106,109)(111,112);(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)(86,94,97,90)(87,98,101,91)(88,93)(89,92)(95,105)(96,110,102,106)(99,107)(100,111,103,108)(104,109)(112,113)(114,115)

Polycyclic presentation:

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

Conjugacy classes & character table

1768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 76389);