#34204017

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#362×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #12751

Subgroups

	subgroup	G/H
center	C₂	#23763329
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#362×C₇₉	C₂
socle	C₂×C₇₉	#1786
prime radical	C₂×#19099376	C₂
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#362	D₇₉
hypocenter	C₇₉	#12751
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 143 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)(80,83,86,102)(81,88,92,113)(82,94,97,125)(84,100,107,135)(85,103,108,101)(87,109,111,139)(89,112,116,132)(90,117,91,99)(93,119,121,142)(95,123,127,143)(96,129,98,124)(104,131,115,140)(105,133,128,137)(106,136,126,134)(110,138,118,130)(114,122,120,141);(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)(80,82)(81,87)(83,94)(84,105)(85,96)(86,97)(88,109)(89,114)(90,110)(91,118)(92,111)(93,115)(95,126)(98,108)(99,130)(100,133)(101,124)(102,125)(103,129)(104,121)(106,127)(107,128)(112,122)(113,139)(116,120)(117,138)(119,140)(123,134)(131,142)(132,141)(135,137)(136,143);(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)(82,93,97,121)(83,99,102,117)(84,104)(85,90)(86,92)(87,95,111,127)(88,101,113,103)(89,106)(91,108)(94,122)(96,120,98,114)(100,132,135,112)(105,118,128,110)(107,115)(109,137)(116,126)(119,129)(123,138)(124,142)(125,141)(130,143)(131,134,140,136)(133,139)

Polycyclic presentation:

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

Conjugacy classes & character table

1288 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 85316);