C₂×#365×C₁₀₉ #28359408

order	13952
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#365×C₁₀₉

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#106
commutator	C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

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

normal subgroups only

This group has 152 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) = 127
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 127 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄×#365×C₁₀₉, C₁₀₉×#12677, C₁₀₉×#12831, …
maximal normal subgroup of	C₄×#365×C₁₀₉, C₁₀₉×#12677, C₁₀₉×#12831, …
maximal quotient of	C₂×C₁₀₉×#1368, C₁₀₉×#12000, C₁₀₉×#12019, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 886);