#34209308

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #13737

Subgroups

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₇₉
commutator, socle	C₂×C₇₉	C₂²×C₃₂
Fitting	C₇₉×#2258	C₂
prime radical	#41×C₇₉	C₁₆
cosocle	C₃₂×C₇₉	C₂³
hypercenter	#2258	D₇₉
hypocenter	C₇₉	#13737
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃₂ ◁ #2258

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 207
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 207 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)(82,94)(88,109)(95,124)(96,125)(97,126)(98,127)(110,144)(111,145)(112,146)(113,147)(128,164)(129,165)(130,166)(131,167)(132,168)(133,169)(148,179)(149,180)(150,181)(151,182)(152,183)(153,184)(170,194)(171,195)(172,196)(173,197)(185,200)(186,201)(187,202)(188,203)(198,206)(204,207);(80,82)(81,88)(83,95)(84,96)(85,97)(86,98)(87,94)(89,110)(90,111)(91,112)(92,113)(93,109)(99,128)(100,129)(101,130)(102,124)(103,131)(104,132)(105,125)(106,133)(107,126)(108,127)(114,148)(115,149)(116,150)(117,144)(118,151)(119,152)(120,145)(121,153)(122,146)(123,147)(134,170)(135,171)(136,164)(137,172)(138,165)(139,166)(140,173)(141,167)(142,168)(143,169)(154,185)(155,186)(156,179)(157,187)(158,180)(159,181)(160,188)(161,182)(162,183)(163,184)(174,198)(175,194)(176,195)(177,196)(178,197)(189,204)(190,200)(191,201)(192,202)(193,203)(199,206)(205,207);(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)(80,81,83,89,84,90,99,114,85,91,100,115,103,118,134,154,86,92,101,116,104,119,135,155,106,121,137,157,140,160,174,189,87,93,102,117,105,120,136,156,107,122,138,158,141,161,175,190,108,123,139,159,142,162,176,191,143,163,177,192,178,193,199,205)(82,88,95,110,96,111,128,148,97,112,129,149,131,151,170,185,98,113,130,150,132,152,171,186,133,153,172,187,173,188,198,204,94,109,124,144,125,145,164,179,126,146,165,180,167,182,194,200,127,147,166,181,168,183,195,201,169,184,196,202,197,203,206,207)

Polycyclic presentation:

2^8,79|a^2=d,d^2=e,e^2=f,f^2=g,g^2=h,i^a=i^78,c^b=ch

Conjugacy classes & character table

3280 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90607);