C₁₄₉×#7605 #49186554

order	38144
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₄₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₄₉×#7605

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₄₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₄₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₄₉
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₄₉ ◁ C₄³×C₁₄₉ ◁ G

440 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 38144

A minimal faithful permutation representation — 3 generators on 181 points:

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

Conjugacy classes & character table

7748 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 593);