C₁₀₉×#1379 #28358633

order	13952
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#1379

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₀₉	D₄
commutator	C₄	C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂²×C₁₀₉
socle	C₂²×C₁₀₉	C₂²⋊C₈
prime radical	C₂³×C₁₀₉	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

136 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 2 generators on 145 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₉×#1379, C₁₀₉×#12554, C₁₀₉×#12597, …
maximal normal subgroup of	C₂×C₁₀₉×#1379, C₁₀₉×#12554, C₁₀₉×#12597, …
maximal quotient of	C₁₀₉×#7141, C₁₀₉×#7144, C₁₀₉×#7147, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 111);