C₁₄₉×#19008 #49197957

order	38144
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₄₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₄₉×#19008

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₄₉
prime radical	C₂³×C₁₄₉	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

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

normal subgroups only

This group has 786 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) = 197
transitive degree	μ_t(G) = 19072

A minimal faithful permutation representation — 4 generators on 197 points:

(1,2,6,3)(4,11,12,16)(5,10,15,8)(7,14,9,13)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,31)(25,34)(30,38)(32,41)(35,42)(36,39)(37,40)(43,45)(44,47)(46,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197);(1,3,6,2)(4,9,12,7)(5,8,15,10)(11,14,16,13)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,39)(27,41)(28,29)(33,47)(34,45)(35,46)(37,44)(40,48)(42,43);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,11)(13,15)(17,20,23,35)(18,25,26,40)(19,30,32,46)(21,34,22,37)(24,38,39,48)(27,33,28,42)(29,43,41,47)(31,44,36,45);(18,26)(24,39)(25,40)(27,28)(29,41)(33,42)(38,48)(43,47)

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, 11996);