C₄×#338×C₁₀₉ #42128692

order	27904
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₄³×C₁₀₉ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₄×#338×C₁₀₉

Subgroups

	subgroup	G/H
center	C₄²×C₁₀₉	C₄²
commutator	C₄	C₄³×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#95
prime radical	C₂⁴×C₁₀₉	C₂×Q₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₉ ◁ G

858 conjugacy classes of subgroups; only the 454 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 454 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:

(1,2,3,4)(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);(5,6,8,12,10,13,9,18)(7,14,16,11,17,19,15,20);(5,7,9,16,10,17,8,15)(6,11,12,14,13,20,18,19)

Conjugacy classes & character table

9592 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 3686);