C₂×#20266068 #27187346

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

Aliases

direct product

C₂×#20266068

Splittings

Fitting	C₂×C₁₆×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ C₂×#347

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×#87938
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₈	C₂²×#87938
Fitting	C₂×C₁₆×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#97
prime radical	C₂²×D₁₀₁	M₁₆
cosocle	C₈×D₁₀₁	C₂³
hypercenter	C₂×C₁₆	#87938
hypocenter	C₁₀₁	C₂×#347
LCS	G ▷ C₄×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

168 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 82 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:

(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)(32,47,48,64)(36,51,49,60)(40,53,54,56)(44,57,58,52)(46,61,62,50)(55,68,69,75)(59,72,73,71)(63,76,77,67)(65,70,79,66)(74,83,84,86)(78,87,88,82)(80,85,90,81)(89,94,95,93)(91,96,97,92)(98,100,101,99)(105,109,111,112)(106,113)(110,118,119,117)(114,116);(1,2,5,10,19,32,49,35,22,13,6,11,20,33,50,66,69,54,39,26,15,24,37,52,67,81,84,73,58,43,28,41,56,71,82,92,95,88,77,62,45,60,75,86,93,99,101,97,90,79,64,47,65,80,91,98,100,94,83,68,51,30,46,63,78,89,96,87,72,53,34,17,29,44,59,74,85,76,57,38,21,8,16,27,40,55,70,61,42,25,12,3,7,14,23,36,48,31,18,9,4)(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, 20724);