C₁₀₇×#7294 #41033258

order	27392
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₇×#7294

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#340
commutator	C₂×C₈	C₄²×C₁₀₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₀₇
prime radical	C₂³×C₁₀₇	#106
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 139
transitive degree	μ_t(G) = 13696

A minimal faithful permutation representation — 2 generators on 139 points:

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

Conjugacy classes & character table

4922 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 282);