C₁₄₉×#7502 #49186451

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₄₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₄₉×#7502

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	#354
commutator	C₄²	C₄²×C₁₄₉
Frattini, cosocle	C₂×#124	C₂²×C₁₄₉
prime radical	C₂²×D₄×C₁₄₉	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂³×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ C₂×#124×C₁₄₉ ◁ G

286 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 185
transitive degree	μ_t(G) = 9536

A minimal faithful permutation representation — 2 generators on 185 points:

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

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, 490);