Q₁₆×C₁₄₉ #14558705

order	2384
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₄₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

Q₁₆×C₁₄₉

Subgroups

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

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 165
transitive degree	μ_t(G) = 2384

A minimal faithful permutation representation — 2 generators on 165 points:

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

Conjugacy classes & character table

1043 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	SD₃₂×C₁₄₉, Q₃₂×C₁₄₉, C₂×Q₁₆×C₁₄₉, …
maximal normal subgroup of	SD₃₂×C₁₄₉, Q₃₂×C₁₄₉, C₂×Q₁₆×C₁₄₉, …
maximal quotient of	#103×C₁₄₉, #107×C₁₄₉, C₂×Q₁₆×C₁₄₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2384, 9);

Q16×C149 #14558705