C₁₀₇×#7083 #41033047

order	27392
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₀₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₇×#7083

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#327
commutator	C₂²×C₈	C₂×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₁₀₇
prime radical	C₂³×C₁₀₇	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ C₂×C₄⋊C₈×C₁₀₇ ◁ G

172 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 6848

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

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

Conjugacy classes & character table

3638 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 71);