#27186795

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #1728

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87938
commutator	C₂²×C₁₀₁	C₂×C₄²
Fitting	C₂³×C₄×C₁₀₁	C₄
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂³×C₄	#87938
hypocenter	C₁₀₁	#1728
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

480 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 117
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 117 points:

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

Polycyclic presentation:

2^7,101|a^2=e,b^2=f,e^2=g,b^a=bd,c^a=cg,h^a=h^10,h^e=h^100

Conjugacy classes & character table

856 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20173);