C₁₀₉×#2115 #28359369

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

Aliases

direct product

C₁₀₉×#2115

Subgroups

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#7207, C₁₀₉×#7208, C₁₀₉×#11483, …
maximal normal subgroup of	C₁₀₉×#7207, C₁₀₉×#7208, C₁₀₉×#11483, …
maximal quotient of	C₁₀₉×#11419, C₁₀₉×#11441, C₁₀₉×#11461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 847);