C₁₀₇×#7445 #41033409

order	27392
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₀₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₇×#7445

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₀₇	#357
commutator	C₂×C₁₆	C₂×C₄×C₁₀₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₀₇
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₄²×C₁₀₇ ◁ C₄×C₈×C₁₀₇ ◁ C₄×C₁₆×C₁₀₇ ◁ G

132 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 147
transitive degree	μ_t(G) = 27392

A minimal faithful permutation representation — 2 generators on 147 points:

(1,2,3,6,4,5,7,8)(10,15)(11,19)(12,21)(13,25)(16,31)(17,34)(18,30)(20,24)(22,38)(23,28)(26,32)(27,29)(33,39)(35,36)(37,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(2,5)(3,7)(9,10,14,18)(11,20,23,37)(12,24,26,40)(13,27,25,17)(15,28,30,19)(16,32,33,21)(22,39,35,31)(29,36,34,38)

Conjugacy classes & character table

5243 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 433);