C₂×#20266069 #27187348

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#20266069

Splittings

Sylow

C₁₀₁ ⋊ C₂×#347

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×#87938
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂×C₄	C₂²×#87938
Fitting	C₂×C₄²×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#97
prime radical	C₂³×C₁₀₁	M₁₆
cosocle	C₂×#11717579	C₂³
hypercenter	C₂×C₄²	#87938
hypocenter	C₁₀₁	C₂×#347
LCS	G ▷ C₄×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

168 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 119
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 119 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)(105,109,111,112)(106,113)(110,118,119,117)(114,116);(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)(104,105,106,110,107,112,114,118,108,111,113,119,115,109,116,117);(102,103)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20726);