C₁₀₉×#1757 #28359011

order	13952
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₀₉×#1757

Subgroups

	subgroup	G/H
center	C₈×C₁₀₉	C₂²⋊C₄
commutator	C₂²	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂×C₁₀₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

284 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#7207, C₁₀₉×#7208, C₁₀₉×#12613, …
maximal normal subgroup of	C₁₀₉×#7207, C₁₀₉×#7208, C₁₀₉×#12613, …
maximal quotient of	C₁₀₉×#8156, C₁₀₉×#8157, C₁₀₉×#8158, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 489);