#34180619

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×#1473 ⋊ C₂
Sylow	C₇₉ ⋊ #8490

Subgroups

	subgroup	G/H
center	C₂²	#19098418
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1473	C₂
socle	C₂²×C₇₉	#379
prime radical	#19099151	C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1473	D₇₉
hypocenter	C₇₉	#8490
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1473

630 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)(81,87)(82,91)(84,98)(85,95)(86,100)(88,94)(89,105)(90,106)(92,107)(93,109)(96,99)(97,108)(101,111)(102,103)(104,110)(112,115)(113,119)(114,123)(116,127)(117,128)(118,132)(120,134)(121,126)(122,136)(124,138)(125,139)(129,141)(130,142)(131,143)(133,140)(135,137);(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)(80,82,85,92)(81,86,88,90)(83,91,95,107)(84,97,99,93)(87,100,94,106)(89,104,103,101)(96,109,98,108)(102,111,105,110)(112,114,117,125)(113,118,121,133)(115,123,128,139)(116,130,131,124)(119,132,126,140)(120,135,129,122)(127,142,143,138)(134,137,141,136);(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)(80,81,84,89,85,88,99,103)(82,90,93,104,92,86,97,101)(83,94,96,105,95,87,98,102)(91,106,108,111,107,100,109,110)(112,113)(114,122)(115,126,128,119)(116,129)(117,121)(118,124)(120,131)(123,137,139,136)(125,135)(127,134,143,141)(130,133)(132,138,140,142)

Polycyclic presentation:

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

Conjugacy classes & character table

1285 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 61918);