#34205356

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #13016

Subgroups

	subgroup	G/H
center	C₄	#19098695
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#508×C₇₉	C₂
socle	C₂×C₇₉	#1879
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#508	D₇₉
hypocenter	C₇₉	#13016
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

462 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,83)(81,90)(82,96)(84,107)(85,112)(86,103)(87,114)(88,106)(89,115)(91,102)(92,119)(93,127)(94,122)(95,128)(97,132)(98,131)(99,135)(100,136)(101,134)(104,109)(105,108)(110,138)(111,139)(113,133)(116,143)(117,130)(118,142)(120,125)(121,124)(123,137)(126,141)(129,140);(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)(80,82,87,113,88,101,108,100)(81,89,93,126,94,118,124,117)(83,96,114,133,106,134,105,136)(84,98,86,97,109,99,85,111)(90,115,127,141,122,142,121,130)(91,123,92,95,125,116,110,129)(102,137,119,128,120,143,138,140)(103,132,104,135,112,139,107,131);(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)(80,81,86,92,87,93,109,125,88,94,85,110,108,124,84,91)(82,95,99,118,100,123,97,126,101,129,98,89,113,116,111,117)(83,102,104,127,105,138,103,90,106,120,107,121,114,119,112,122)(96,130,132,137,133,115,135,128,134,141,139,143,136,142,131,140)

Polycyclic presentation:

2^8,79|a^2=f,b^2=dfg,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^78,d^b=dg,e^b=eg,f^b=fg,g^b=gh,i^b=i^78,d^c=dg,e^c=egh,f^c=fg,g^c=gh

Conjugacy classes & character table

1762 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86655);