C₁₄₉×#7387 #49186336

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₄₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₄₉×#7387

Subgroups

	subgroup	G/H
center	C₈×C₁₄₉	D₁₆
commutator	C₁₆	C₂×C₈×C₁₄₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₄₉
socle	C₂×C₁₄₉	#1331
prime radical	#508×C₁₄₉	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₄₉ ◁ C₂×C₈×C₁₄₉ ◁ C₄×C₈×C₁₄₉ ◁ C₄×C₁₆×C₁₄₉ ◁ G

120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 213
transitive degree	μ_t(G) = 9536

A minimal faithful permutation representation — 2 generators on 213 points:

(1,2,5,11,7,12,27,40,8,14,28,41,31,13,6,29)(3,15,17,43,18,44,49,63,20,45,50,64,19,51,30,36)(4,21,23,37,25,52,58,39,26,54,59,38,60,53,24,10)(9,22,32,46,34,55,61,48,35,57,62,47,42,56,33,16)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3)(2,9)(4,22)(5,17)(6,30)(7,18)(8,20)(10,36)(11,32)(12,34)(13,42)(14,35)(15,21)(16,24)(19,31)(23,46)(25,55)(26,57)(27,49)(28,50)(29,33)(37,43)(38,64)(39,63)(40,61)(41,62)(44,52)(45,54)(47,59)(48,58)(51,53)(56,60)

Conjugacy classes & character table

11324 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 375);