C₁₄₉×#12971 #49191920

order	38144
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₄₉×#12971

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₄₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₄₉
prime radical	C₂³×C₁₄₉	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂³×C₁₄₉ ◁ C₂²×Q₈×C₁₄₉ ◁ G

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

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 185 points:

(1,2,4,3)(5,6)(7,17)(8,14)(9,22)(10,25)(11,16)(12,18)(13,21)(15,27)(19,35)(20,32)(23,30)(24,31)(26,28)(29,36)(33,34)(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);(1,3,4,2)(5,7)(6,12)(8,18)(9,23)(10,26)(11,20)(13,28)(14,32)(15,29)(16,30)(17,22)(19,27)(21,36)(24,34)(25,33)(31,35);(5,8,11,9)(6,13,16,31)(7,18,20,23)(10,24,27,21)(12,28,30,35)(14,15,22,25)(17,33,32,29)(19,36,26,34)

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, 5959);