C₁₄₉×#7382 #49186331

order	38144
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₄₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₄₉×#7382

Subgroups

	subgroup	G/H
center	C₈×C₁₄₉	#102
commutator	C₈	C₂×C₁₆×C₁₄₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₄₉
socle	C₂×C₁₄₉	D₄⋊C₁₆
prime radical	#135×C₁₄₉	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₄₉ ◁ C₂×C₁₆×C₁₄₉ ◁ C₄×C₁₆×C₁₄₉ ◁ G

118 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 58 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:

(1,2,5,12,7,11,28,39,8,13,29,42,4,21,23,51,9,14,30,43,24,40,54,63,6,31,27,58,25,44,55,64)(3,15,17,45,19,10,33,34,18,36,50,60,22,37,52,61,20,47,26,56,16,38,48,41,32,46,57,62,49,35,53,59)(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)(2,10)(4,22)(5,26)(6,32)(7,19)(8,18)(9,20)(11,36)(12,41)(13,37)(14,38)(15,44)(16,24)(17,30)(21,47)(23,53)(25,49)(27,50)(28,48)(29,57)(31,35)(33,54)(34,43)(39,62)(40,46)(42,59)(45,51)(52,55)(56,64)(58,61)(60,63)

Conjugacy classes & character table

9536 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 370);