C₁₀₉×#2113 #28359367

order	13952
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#2113

Subgroups

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

228 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 120 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)(5,6,8,12,9,13,18,24,10,14,19,25,20,26,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(2,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,22)(13,23)(14,21)(18,29)(19,27)(20,28)(24,33)(25,31)(26,32)(30,35)(34,36);(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36)

Conjugacy classes & character table

6104 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7207, C₁₀₉×#7208, C₁₀₉×#7227, …
maximal normal subgroup of	C₁₀₉×#7207, C₁₀₉×#7208, C₁₀₉×#7227, …
maximal quotient of	C₁₀₉×#11419, C₁₀₉×#11441, C₁₀₉×#11461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 845);