C₂×#23761617 #34209318

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#23761617

Splittings

Sylow

C₇₉ ⋊ C₂×Q₁₂₈

Subgroups

	subgroup	G/H
center	C₂²	#19098398
commutator, cosocle	C₃₂×C₇₉	C₂³
Frattini	C₃₂	C₂²×D₇₉
Fitting	C₂×Q₆₄×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	D₃₂
hypercenter	C₂×Q₆₄	D₇₉
hypocenter	C₇₉	C₂×Q₁₂₈
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 209
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 209 points:

(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)(82,84,89,101)(83,90,95,124)(85,103,108,156)(86,110,113,173)(87,115,117,177)(88,119,116,100)(91,126,129,190)(92,131,133,198)(93,135,136,200)(94,137,118,123)(96,138,141,102)(97,125,120,105)(98,130,146,104)(99,134,148,111)(106,163,165,209)(107,167,164,155)(109,121,170,183)(112,175,158,172)(114,122,176,185)(127,192,193,206)(128,194,178,189)(132,199,168,197)(139,203,204,162)(140,152,149,166)(142,191,180,208)(143,161,179,188)(144,201,205,150)(145,159,171,195)(147,202,207,151)(153,187,184,157)(154,186,181,160)(169,196,174,182);(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)(82,83,89,95)(84,96,101,141)(85,102,108,138)(86,109,113,170)(87,114,117,176)(88,118,116,94)(90,120,124,97)(91,125,129,105)(92,130,133,104)(93,134,136,111)(98,144,146,205)(99,147,148,207)(100,149,119,140)(103,131,156,198)(106,162,165,203)(107,166,164,152)(110,126,173,190)(112,174,158,169)(115,139,177,204)(121,181,183,154)(122,184,185,153)(123,143,137,179)(127,191,193,208)(128,161,178,188)(132,159,168,195)(135,180,200,142)(145,206,171,192)(150,187,201,157)(151,186,202,160)(155,199,167,197)(163,196,209,182)(172,194,175,189);(80,81)

Conjugacy classes & character table

1552 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90617);

C2×#23761617 #34209318