C₁₀₉×#2211 #28359465

order	13952
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#2211

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

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

normal subgroups only

This group has 158 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) = 145
transitive degree	μ_t(G) = 6976

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

(2,4)(5,6)(7,15)(8,11)(9,21)(10,24)(12,17)(13,23)(14,20)(16,31)(18,35)(19,32)(22,30)(25,29)(26,33)(27,34)(28,36)(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);(1,2,3,4)(6,11)(12,26)(13,27)(14,24)(15,31)(21,32)(28,30)(29,35);(5,7,8,16)(6,12,11,26)(9,22,19,36)(10,25,20,18)(13,29,27,35)(14,30,24,28)(15,32,31,21)(17,34,33,23)

Conjugacy classes & character table

3161 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₉×#2211, C₁₀₉×#33579, C₁₀₉×#33581, …
maximal normal subgroup of	C₂×C₁₀₉×#2211, C₁₀₉×#33579, C₁₀₉×#33581, …
maximal quotient of	C₁₀₉×#12419, C₁₀₉×#12423, C₁₀₉×#12429, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 943);