C₁₀₉×#32768 #42150762

order	27904
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₁₀₉×#32768

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×#117×C₁₀₉ ◁ G

736 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 157
transitive degree	μ_t(G) = 13952

A minimal faithful permutation representation — 4 generators on 157 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,38)(29,37)(30,39)(31,41)(32,40)(33,43)(34,44)(35,42)(45,47)(46,48)(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);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,25,37)(20,30,32,46)(21,33,35,28)(24,39,40,48)(26,43,42,38)(27,34,45,31)(36,44,47,41);(3,7)(6,8)(10,14)(13,16)(17,20,22,32)(18,24,25,40)(19,27,29,45)(21,34,35,31)(23,36,37,47)(26,44,42,41)(28,46,33,30)(38,48,43,39);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,25)(23,37)(24,40)(26,42)(36,47)(38,43)(39,48)(41,44)

Conjugacy classes & character table

3706 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 25756);