C₁₄₉×#18997 #49197946

order	38144
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₄₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₄₉×#18997

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₄₉
prime radical	C₂×H(3,𝔽₄)×C₁₄₉	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 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:

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

Conjugacy classes & character table

5066 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 11985);