C₁₀₇×#7787 #41033751

order	27392
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₇×#7787

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
prime radical	C₂²×D₄×C₁₀₇	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

430 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 3 generators on 139 points:

(2,5)(3,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(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);(1,2,4,6)(3,8,7,5)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(10,13)(11,16)(14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)

Conjugacy classes & character table

3959 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 775);