#34203988

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×#350×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ #12747

Subgroups

	subgroup	G/H
center	C₄	#19098699
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#350×C₇₉	C₂
socle	C₂×C₇₉	#1786
prime radical	#23778174	C₂
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#350	D₇₉
hypocenter	C₇₉	#12747
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#350

886 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)(81,89)(82,94)(84,99)(85,107)(86,100)(87,101)(88,110)(90,114)(91,116)(92,115)(93,119)(95,123)(96,126)(97,125)(98,108)(102,133)(103,130)(104,136)(105,131)(106,132)(109,124)(111,139)(112,128)(113,138)(117,122)(118,140)(120,142)(121,141)(127,143)(129,134)(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)(81,88)(83,94)(84,103)(85,96)(86,109)(87,97)(89,110)(90,117)(91,111)(92,113)(93,102)(95,105)(98,128)(99,130)(100,124)(101,125)(104,127)(106,135)(107,126)(108,112)(114,122)(115,138)(116,139)(118,120)(119,133)(121,134)(123,131)(129,141)(132,137)(136,143)(140,142);(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)(83,98,101,116)(84,102)(85,91)(86,108)(87,92)(88,95)(89,100,115,107)(90,105)(94,122,125,142)(96,120)(97,121)(99,114,132,140)(103,111)(104,118)(106,134)(109,117)(110,137,138,130)(112,135)(113,127)(119,124,141,126)(123,128,143,139)(129,136,133,131)

Polycyclic presentation:

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

Conjugacy classes & character table

2230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 85287);