C₁₀₉×#2141 #28359395

order	13952
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#2141

Subgroups

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

220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 104 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)(7,16)(8,13)(9,22)(10,26)(11,15)(12,18)(14,24)(17,30)(19,36)(20,33)(21,31)(23,29)(25,32)(27,34)(28,35)(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);(5,7,11,20)(6,12,15,28)(8,19,10,17)(9,23,25,34)(13,29,26,27)(14,30,31,36)(16,32,33,22)(18,21,35,24);(5,8,11,10)(6,13,15,26)(7,17,20,19)(9,21,25,24)(12,27,28,29)(14,22,31,32)(16,30,33,36)(18,34,35,23)

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12648, C₁₀₉×#12742, C₁₀₉×#12751, …
maximal normal subgroup of	C₁₀₉×#12648, C₁₀₉×#12742, C₁₀₉×#12751, …
maximal quotient of	C₁₀₉×#11594, C₁₀₉×#11650, C₁₀₉×#11662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 873);