C₁₀₇×#7466 #41033430

order	27392
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₁₀₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₇×#7466

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₀₇	C₄²
commutator	C₄	C₄×C₁₆×C₁₀₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₁₀₇
socle	C₂²×C₁₀₇	#346
prime radical	C₂³×C₁₀₇	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₇ ◁ G

182 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 143
transitive degree	μ_t(G) = 6848

A minimal faithful permutation representation — 2 generators on 143 points:

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

Conjugacy classes & character table

9416 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 454);