C₁₄₉×#13570 #49192519

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₄₉×#13570

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₁₄₉
Frattini, cosocle	#124	C₂³×C₁₄₉
prime radical	#3018×C₁₄₉	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ #532×C₁₄₉ ◁ G

482 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 181 points:

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

Conjugacy classes & character table

4172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 6558);