C₁₀₃×#2135 #27191501

order	13184
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₃×#2135

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₃	C₂×#99
commutator	C₂×C₄	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₂×Q₈	C₂³×C₁₀₃
prime radical	C₂×#41×C₁₀₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₃ ◁ C₂×C₄×C₁₀₃ ◁ C₂×#41×C₁₀₃ ◁ G

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Conjugacy classes & character table

2369 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₃×#12712, C₁₀₃×#12713, C₁₀₃×#12721, …
maximal normal subgroup of	C₁₀₃×#12712, C₁₀₃×#12713, C₁₀₃×#12721, …
maximal quotient of	C₁₀₃×#11434, C₁₀₃×#11450, C₁₀₃×#11481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 867);