C₁₁₃×#1461 #28386257

order	14464
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₁₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₁₃×#1461

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₁₃	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₁₁₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₁₃
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₁₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁₃ ◁ C₂×C₄²×C₁₁₃ ◁ G

248 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 137
transitive degree	μ_t(G) = 7232

A minimal faithful permutation representation — 3 generators on 137 points:

(1,2,4,7,5,8,3,6)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(1,3,5,4)(2,6,8,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)

Conjugacy classes & character table

4972 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₁₃×#7206, C₁₁₃×#7286, C₈×#101×C₁₁₃, …
maximal normal subgroup of	C₁₁₃×#7206, C₁₁₃×#7286, C₈×#101×C₁₁₃, …
maximal quotient of	C₁₁₃×#7568, C₁₁₃×#7570, C₁₁₃×#7585, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14464, 193);