#34204726

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12911

Subgroups

	subgroup	G/H
center	C₄	#19098600
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#368×C₇₉	C₂
socle	C₂×C₇₉	#1846
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#368	D₇₉
hypocenter	C₇₉	#12911
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

266 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)(81,89,92,113)(82,94)(84,100)(85,101)(86,103)(87,102)(88,109,111,137)(90,112,114,130)(91,99,108,129)(93,115,117,138)(95,119)(96,120)(97,122)(98,121)(104,128,134,143)(105,131)(106,133)(107,132)(110,127,135,140)(116,118,126,139)(123,136)(124,142)(125,141);(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)(80,82,84,95,86,98,106,125,87,97,107,124,85,96,105,123)(81,88,90,93,91,110,104,116,92,111,114,117,108,135,134,126)(83,94,100,119,103,121,133,141,102,122,132,142,101,120,131,136)(89,109,112,115,99,127,128,118,113,137,130,138,129,140,143,139);(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)(82,93,97,117)(83,99,102,129)(84,104)(85,91)(86,108)(87,92)(88,95,111,124)(89,101,113,103)(90,106)(94,118)(96,116,98,126)(100,130,132,112)(105,114)(107,134)(109,136)(110,123,135,125)(115,121)(119,140)(120,138)(122,139)(127,142)(128,131,143,133)(137,141)

Polycyclic presentation:

2^8,79|a^2=f,b^2=d,c^2=eg,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,f^b=fh,i^b=i^78,f^c=fh,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, 86025);