C₁₀₉×#2143 #28359397

order	13952
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#2143

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	C₂×D₈
commutator	C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	C₂×#102
prime radical	C₂×#135×C₁₀₉	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#12649, C₁₀₉×#12651, C₁₀₉×#12740, …
maximal normal subgroup of	C₁₀₉×#12649, C₁₀₉×#12651, C₁₀₉×#12740, …
maximal quotient of	C₁₀₉×#11592, C₁₀₉×#11593, C₁₀₉×#11594, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 875);