C₂×#352×C₁₀₉ #28359373

order	13952
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#352×C₁₀₉

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#99
commutator	C₂³	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×D₄	C₂³×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₁₀₉ ◁ C₂²×D₄×C₁₀₉ ◁ G

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

normal subgroups only

This group has 278 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) = 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);(3,4)(5,11)(6,14,15,9)(7,10)(8,12,16,18)(13,17);(3,5,7,13)(4,8,10,16)(6,12,15,18)(9,11,14,17)

Conjugacy classes & character table

2834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×#352×C₁₀₉, C₁₀₉×#12621, C₁₀₉×#12688, …
maximal normal subgroup of	C₄×#352×C₁₀₉, C₁₀₉×#12621, C₁₀₉×#12688, …
maximal quotient of	C₂×C₁₀₉×#1317, C₂×C₁₀₉×#1326, C₁₀₉×#11472, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 851);