C₅×#73042 #14510766

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73042

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5402
commutator, cosocle	#131⋊C₃	C₂²×C₅
Frattini	C₈	C₂×C₅×S₄
Fitting	C₅×#504	D₃
prime radical	C₅×#41⋊C₃	D₄
hypercenter	C₅×C₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×D₈
LCS	G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×C₁₆

164 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 133
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 133 points:

(1,2,3,4,5)(6,70,115,59,77,123,31,63,36,26,60,132,28,66,83,22,37,72,21,126,88,23,29,118,13,24,114,93,33,122,9,94,39,8,100,133,10,116,80,41,11,69,46,112,87,62,81,119)(7,91,125,55,85,129,49,56,52,45,74,130,32,96,103,20,38,99,27,120,105,42,47,110,19,43,124,65,50,128,15,71,53,14,82,131,16,106,102,58,17,98,76,108,104,54,92,111)(12,25,61,113,34,67,84,95,40,73,101,127,78,117,30,64)(18,44,75,109,51,97,90,68,35,89,79,121,86,107,48,57);(6,7,13,19)(8,20,26,58)(9,27,31,76)(10,32,28,16)(11,35,37,18)(12,38,40,17)(14,41,45,22)(15,46,49,21)(23,65,62,55)(24,68,70,57)(25,71,73,56)(29,79,81,75)(30,82,84,74)(33,51,77,86)(34,50,78,85)(36,53,39,52)(42,59,54,93)(43,64,91,95)(44,63,89,94)(47,61,92,101)(48,60,90,100)(66,109,116,121)(67,108,117,120)(69,111,72,110)(80,125,83,124)(87,104,88,105)(96,127,106,113)(97,126,107,112)(98,118,99,119)(102,114,103,115)(122,130,123,131)(128,133,129,132)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236464);