#491×C₁₉₉ #27163641

order	12736
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

#491×C₁₉₉

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₉₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₉ ◁ C₄²×C₁₉₉ ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 223
transitive degree	μ_t(G) = 12736

A minimal faithful permutation representation — 3 generators on 223 points:

(1,2)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,23,17)(12,24,22,16)(13,19,14,18)(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);(1,3,4,7)(2,5,6,8)(9,11,13,22,15,23,14,12)(10,16,18,21,20,24,19,17);(2,6)(5,8)(9,12,13,11,15,22,14,23)(10,17,18,16,20,21,19,24)

Conjugacy classes & character table

3184 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#38766885, #38767023, #38768171, …
maximal subgroup of	#1356×C₁₉₉, #1413×C₁₉₉, #1662×C₁₉₉, …
maximal normal subgroup of	#1356×C₁₉₉, #1413×C₁₉₉, #1662×C₁₉₉, …
maximal quotient of	#1836×C₁₉₉, #1935×C₁₉₉, #1960×C₁₉₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12736, 172);

#491×C199 #27163641