#45692811

order	33522
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃₇×C₁₅₁ ⋊ C₂×C₃
Hall	#30660896 ⋊ C₂
Hall	D₅₅₈₇ ⋊ C₃
Sylow	C₁₅₁ ⋊ D₃₇⋊C₃
Sylow	C₃₇ ⋊ D₁₅₁⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₇×C₁₅₁	C₂×C₃
LCS	G ▷ C₃₇×C₁₅₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 188
transitive degree	μ_t(G) = 5587

A minimal faithful permutation representation — 2 generators on 188 points:

(2,6,7,3,4,8)(5,14,15,9,10,23)(11,26,27,17,18,34)(12,25,21,19,16,13)(20,36,37,29,28,32)(22,24,35,31,30,33)(39,43,44,40,41,45)(42,51,52,46,47,61)(48,64,65,54,55,78)(49,63,59,57,53,50)(56,83,84,67,68,101)(58,81,82,70,66,75)(60,62,80,72,73,76)(69,106,107,86,87,130)(71,104,105,89,85,98)(74,79,103,91,92,99)(77,119,117,95,96,94)(88,135,136,109,110,163)(90,133,134,112,108,127)(93,102,132,114,115,128)(97,147,148,120,121,123)(100,153,145,124,125,118)(111,168,169,138,139,187)(113,166,167,141,137,161)(116,131,165,143,140,162)(122,174,175,149,150,157)(126,179,177,154,155,152)(129,182,173,158,159,146)(142,183,178,160,170,185)(144,164,188,171,172,186)(151,180,181,156,176,184);(1,2,5,11,18,28,33,34,31,21,7,6,12,20,30,10,16,23,32,37,15,25,14,24,29,19,4,8,13,22,27,35,36,26,17,9,3)(38,39,42,48,56,69,88,111,140,87,108,121,125,128,130,161,184,175,134,84,103,119,104,83,102,129,160,141,112,89,70,57,41,45,50,60,74,93,116,144,172,110,137,150,155,159,162,163,185,177,148,105,65,80,81,64,79,100,126,156,149,120,95,73,47,53,61,75,94,118,146,169,188,139,170,176,180,183,168,186,187,173,145,117,82,52,63,51,62,77,97,122,151,154,124,92,55,66,76,78,98,123,152,178,136,165,182,179,174,166,135,164,171,143,114,91,72,59,44,43,49,58,71,90,113,142,158,115,68,85,96,99,101,127,157,181,167,107,132,153,147,133,106,131,138,109,86,67,54,46,40)

Polycyclic presentation:

2,3,37,151|c^a=c^36,d^a=d^150,c^b=c^26,d^b=d^32

Conjugacy classes & character table

937 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33522, 23);