C₁₀₉×Dic₃₁ #27214034

order	13516
isoclinism	isoclinic class of D₃₁
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₀₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₀₉×Dic₃₁
unambiguous semidirect product	C₃₁⋊(C₄×C₁₀₉)

Splittings

derived, Sylow	C₃₁ ⋊ C₄×C₁₀₉
Hall	C₃₁×C₁₀₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₀₉	D₃₁
commutator, hypocenter	C₃₁	C₄×C₁₀₉
Frattini	C₂	D₃₁×C₁₀₉
Fitting, socle, prime radical	C₂×C₃₁×C₁₀₉	C₂
cosocle	C₂×C₃₁	C₂×C₁₀₉
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂×C₁₀₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 13516

A minimal faithful permutation representation — 2 generators on 144 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33,34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)

Conjugacy classes & character table

3706 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13516, 2);

C109×Dic31 #27214034