C₁₄₉×#9458 #49188407

order	38144
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₄₉×#9458

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₄₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₄₉
prime radical	C₂×#127×C₁₄₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

536 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 173
transitive degree	μ_t(G) = 9536

A minimal faithful permutation representation — 3 generators on 173 points:

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

Conjugacy classes & character table

5513 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 2446);