C₁₀₉×#2242 #28359496

order	13952
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₀₉×#2242

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
prime radical	C₂×D₈×C₁₀₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₄×C₈×C₁₀₉ ◁ G

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 149
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 149 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,23)(12,17)(13,31)(14,34)(15,35)(16,22)(18,27)(19,24)(20,33)(21,32)(25,37)(26,36)(28,40)(29,39)(30,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149);(2,5)(6,8)(9,11,24,29,15,26,33,30,16,27,14,25,32,28,13,12)(10,17,31,40,21,37,34,18,22,38,20,36,35,39,19,23);(1,3,4,7)(2,6,5,8)(9,12,13,28,32,25,14,27,16,30,33,26,15,29,24,11)(10,18,19,37,35,40,20,17,22,23,34,39,21,36,31,38)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₉×#2242, C₁₀₉×#33697, C₁₀₉×#33702, …
maximal normal subgroup of	C₂×C₁₀₉×#2242, C₁₀₉×#33697, C₁₀₉×#33702, …
maximal quotient of	C₁₀₉×#12889, C₁₀₉×#13147, C₁₀₉×#13148, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 974);