#27209986

order	13188
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived, Hall	C₇×C₁₅₇ ⋊ C₃×C₄
Hall	Dic₁₀₉₉ ⋊ C₃
Hall	C₇×C₁₅₇⋊C₃ ⋊ C₄
Hall	C₁₅₇⋊C₃ ⋊ Dic₇
Sylow	C₁₅₇ ⋊ C₃×Dic₇
Sylow	C₇ ⋊ Dic₁₅₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	(C₁₅₇⋊C₃)⋊D₇
commutator, hypocenter	C₇×C₁₅₇	C₃×C₄
Fitting, socle, cosocle	C₂×C₇×C₁₅₇	C₂×C₃
prime radical	C₂×C₇×C₁₅₇⋊C₃	C₂
LCS	G ▷ C₇×C₁₅₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 168
transitive degree	μ_t(G) = 4396

A minimal faithful permutation representation — 2 generators on 168 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,124)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,107,108,83,84,86)(63,113,105,87,88,81)(74,100,121,76,101,127)(79,94,126,102,103,122)(85,131,132,109,110,117)(89,135,133,114,115,112)(92,140,129,118,119,106)(104,125,144,111,128,148)(116,151,152,136,137,139)(120,154,149,141,142,134)(123,155,138,145,146,130)(143,156,157,150,147,153)(159,160)(161,162)(163,164)(165,166,167,168);(1,2,5,11,19,32,51,74,75,52,33,20,4,8,13,23,37,56,79,104,109,83,58,36,10,16,24,38,57,81,106,130,136,114,87,55,18,29,39,41,61,86,112,134,150,141,118,78,31,48,59,62,64,90,117,139,153,156,145,103,50,71,84,88,91,93,121,144,155,154,151,128,73,100,110,115,119,122,124,99,126,140,135,131,101,98,125,137,142,146,148,127,70,95,113,107,96,69,94,123,147,157,152,132,97,47,66,82,67,46,65,92,120,143,149,133,108,68,28,43,44,27,42,63,89,116,138,129,105,80,45,15,26,14,25,40,60,85,111,102,77,54,35,22,7,6,12,21,34,53,76,72,49,30,17,9,3)(158,159,161,163,164,162,160)

Polycyclic presentation:

2,3,7,157,2|a^2=e,c^a=c^6,d^a=d^156,d^b=d^12

Conjugacy classes & character table

394 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13188, 21);