#1615×C₁₅₇ #34095973

order	20096
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₅₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1615×C₁₅₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₅₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₅₇
prime radical	#121×C₁₅₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₇ ◁ C₂²×C₄×C₁₅₇ ◁ G

208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 5024

A minimal faithful permutation representation — 3 generators on 189 points:

(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27)(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);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Conjugacy classes & character table

3611 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 347);