C₂×#28385668 #43273690

order	28820
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#28385668

Splittings

Fitting, socle	C₂×C₁₁×C₁₃₁ ⋊ C₂×C₅
derived, Hall	C₁₁×C₁₃₁ ⋊ C₂²×C₅
Hall	#20297888 ⋊ C₂²
Hall	C₂×D₁₄₄₁ ⋊ C₅
Sylow	C₁₃₁ ⋊ C₂×AGL(1,𝔽₁₁)
Sylow	C₁₁ ⋊ C₂×D₁₃₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#28385668
commutator, cosocle, hypocenter	C₁₁×C₁₃₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₁₁×C₁₃₁	C₂×C₅
LCS	G ▷ C₁₁×C₁₃₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 2882

A minimal faithful permutation representation — 2 generators on 144 points:

(4,8,9,7,12,5,6,13,10,11)(15,19,20,38,43,16,17,25,27,67)(18,32,33,92,96,22,23,54,56,135)(21,48,49,50,74,29,30,82,84,86)(24,61,62,26,66,35,36,106,40,41)(28,78,79,75,76,45,46,80,44,85)(31,88,89,72,73,51,52,130,42,120)(34,100,101,103,104,58,59,113,60,139)(37,111,112,122,129,64,65,110,87,123)(39,116,117,68,81,69,70,140,47,115)(53,119,107,128,114,71,91,95,136,141)(55,77,137,57,134,93,94,121,97,98)(63,133,125,126,138,108,109,83,143,102)(90,144,142,127,124,132,131,118,105,99);(1,2)(3,4,7,12,6,9,13,8,11,10,5)(14,15,18,24,37,66,114,142,116,92,136,76,82,124,97,54,79,78,122,84,48,83,127,103,58,32,55,95,36,63,110,140,144,52,90,133,81,123,59,102,139,86,67,62,98,119,89,96,73,101,111,72,40,17,21,31,53,69,20,28,47,27,44,77,121,126,143,137,94,75,38,68,45,25,39,71,51,29,19,26,42,65,113,120,135,130,91,134,106,43,74,104,138,100,129,115,109,132,88,131,117,112,108,61,107,93,23,34,60,105,125,30,50,87,46,80,33,57,99,49,85,128,56,70,118,141,41,64,35,22,16)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28820, 44);

C2×#28385668 #43273690