C₁₀₃×#13359 #39918634

order	26368
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₃×#13359

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂³×C₁₀₃
prime radical	C₂×2₋¹⁺⁴×C₁₀₃	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂³×C₁₀₃ ◁ C₂²×Q₈×C₁₀₃ ◁ G

704 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 6592

A minimal faithful permutation representation — 3 generators on 123 points:

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

Conjugacy classes & character table

3193 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 6347);