C₁₀₉×#3415 #28360669

order	13952
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₀₉×#3415

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₁₀₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ #41×C₁₀₉ ◁ #131×C₁₀₉ ◁ G

370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 4 generators on 141 points:

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

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#13669, C₁₀₉×#13671, C₁₀₉×#33854, …
maximal normal subgroup of	C₁₀₉×#13669, C₁₀₉×#13671, C₁₀₉×#33854, …
maximal quotient of	C₁₀₉×#33381, C₁₀₉×#33410, C₁₀₉×#33454, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2147);