C₁₀₉×#2216 #28359470

order	13952
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#2216

Subgroups

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

154 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 145
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 145 points:

(1,2)(3,4)(5,6,10,14)(7,15,19,31)(8,20,22,35)(9,23,21,13)(11,24,26,17)(12,27,28,16)(18,34,33,30)(25,32,29,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(5,7,16,32,9,18,22,24,10,19,27,36,21,33,8,17)(6,11,20,30,13,25,28,31,14,26,35,34,23,29,12,15);(2,4)

Conjugacy classes & character table

4142 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7419, C₁₀₉×#7421, C₁₀₉×#13664, …
maximal normal subgroup of	C₁₀₉×#7419, C₁₀₉×#7421, C₁₀₉×#13664, …
maximal quotient of	C₁₀₉×#12431, C₁₀₉×#12440, C₁₀₉×#12452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 948);