C₁₀₃×#7702 #39912977

order	26368
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₃×#7702

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₃ ◁ C₄³×C₁₀₃ ◁ G

410 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 135
transitive degree	μ_t(G) = 26368

A minimal faithful permutation representation — 3 generators on 135 points:

(1,2,4,8,5,6,3,7)(9,10,13,15,14,11,12,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30)(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);(2,6)(3,4)(9,11,14,10)(12,15,13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)

Conjugacy classes & character table

4738 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 690);