C₁₀₃×#7489 #39912764

order	26368
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₃×#7489

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₁₀₃
Frattini, cosocle	C₂×#124	C₂²×C₁₀₃
prime radical	C₂²×D₄×C₁₀₃	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂³×C₁₀₃ ◁ C₂²×C₄×C₁₀₃ ◁ C₂×#124×C₁₀₃ ◁ G

302 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 123
transitive degree	μ_t(G) = 3296

A minimal faithful permutation representation — 2 generators on 123 points:

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

Conjugacy classes & character table

3502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 477);