#346×C₂₆₉ #31770083

order	17216
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₂₆₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

#346×C₂₆₉

Subgroups

	subgroup	G/H
center	C₂×C₈×C₂₆₉	C₂²
commutator	C₂	C₄×C₈×C₂₆₉
Frattini, cosocle	C₂×C₈	C₂²×C₂₆₉
socle, prime radical	C₂²×C₂₆₉	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₆₉ ◁ G

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 289
transitive degree	μ_t(G) = 17216

A minimal faithful permutation representation — 2 generators on 289 points:

(5,6,7,11,8,12,13,18,9,10,14,16,15,17,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289);(1,2,3,4)(6,10)(11,16)(12,17)(18,20)

Conjugacy classes & character table

10760 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#46808805, #46808807, #46808844, …
maximal subgroup of	#1313×C₂₆₉, #1330×C₂₆₉, #1333×C₂₆₉, …
maximal normal subgroup of	#1313×C₂₆₉, #1330×C₂₆₉, #1333×C₂₆₉, …
maximal quotient of	#1311×C₂₆₉, #1312×C₂₆₉, #1374×C₂₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17216, 27);

#346×C269 #31770083