C₁₀₉×#12632 #42130626

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₀₉×#12632

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×C₄×Q₈×C₁₀₉ ◁ G

512 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 129
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 129 points:

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

Conjugacy classes & character table

5014 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 5620);