C₁₀₉×#2117 #28359371

order	13952
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#2117

Subgroups

	subgroup	G/H
center	C₈×C₁₀₉	C₂×D₄
commutator	C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₀₉	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 3 generators on 141 points:

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

Conjugacy classes & character table

4796 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12609, C₁₀₉×#12618, C₁₀₉×#12685, …
maximal normal subgroup of	C₁₀₉×#12609, C₁₀₉×#12618, C₁₀₉×#12685, …
maximal quotient of	C₁₀₉×#11421, C₁₀₉×#11422, C₁₀₉×#11443, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 849);