C₃×C₄×C₁₉×C₁₃₇ #44530313

order	31236
nilpotency class	1
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₁₂₃₆
abelian	C₃×C₄×C₁₉×C₁₃₇
multiplicative	(ℤ/31237)ˣ, (ℤ/62474)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃×C₁₉×C₁₃₇
socle, prime radical	C₂×C₃×C₁₉×C₁₃₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 163
transitive degree	μ_t(G) = 31236

A minimal faithful permutation representation — 1 generator on 163 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163)

Conjugacy classes & character table

31236 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	Dic₁₅₆₁₈, D₃₁₂₃₆
maximal subgroup of	Dic₁₅₆₁₈, D₃₁₂₃₆
maximal normal subgroup of	Dic₁₅₆₁₈, D₃₁₂₃₆
automorphism group of	C₃₁₂₃₇
outer automorphism group of	C₃₁₂₃₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31236, 1);

C3×C4×C19×C137 #44530313