#3416×C₁₄₉ #34042329

order	19072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₄₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

#3416×C₁₄₉

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₁₄₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ #41×C₁₄₉ ◁ #131×C₁₄₉ ◁ G

360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 181
transitive degree	μ_t(G) = 4768

A minimal faithful permutation representation — 4 generators on 181 points:

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

Conjugacy classes & character table

4768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19072, 2148);