C₁₀₇×#7266 #41033230

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

Aliases

direct product

C₁₀₇×#7266

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#329
commutator	C₄²	C₄²×C₁₀₇
Frattini, cosocle	C₂×#97	C₂²×C₁₀₇
prime radical	C₂³×C₁₀₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ C₂×#97×C₁₀₇ ◁ G

218 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 127
transitive degree	μ_t(G) = 6848

A minimal faithful permutation representation — 2 generators on 127 points:

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

Conjugacy classes & character table

4280 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 254);