C₁₀₃×#2178 #27191544

order	13184
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₃×#2178

Subgroups

	subgroup	G/H
center	C₈×C₁₀₃	D₈
commutator	C₈	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₃
socle	C₂×C₁₀₃	C₄×D₈
prime radical	#508×C₁₀₃	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₃ ◁ C₂×C₈×C₁₀₃ ◁ C₄×C₈×C₁₀₃ ◁ G

146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 3296

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

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

Conjugacy classes & character table

4532 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₃×#7387, C₁₀₃×#7388, C₁₀₃×#12046, …
maximal normal subgroup of	C₁₀₃×#7387, C₁₀₃×#7388, C₁₀₃×#12046, …
maximal quotient of	C₈×D₁₆×C₁₀₃, C₈×SD₃₂×C₁₀₃, C₈×Q₃₂×C₁₀₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 910);