C₅×D₁₇×D₁₀₉ #49150385

order	37060
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₁₇×D₁₀₉
unambiguous semidirect product	(D₁₇×C₁₀₉)⋊(C₂×C₅)
unambiguous semidirect product	(C₁₇×D₁₀₉)⋊(C₂×C₅)
unambiguous semidirect product	D₁₈₅₃⋊(C₂×C₅)

Splittings

Fitting, socle, Hall	C₅×C₁₇×C₁₀₉ ⋊ C₂²
derived, Hall	C₁₇×C₁₀₉ ⋊ C₂²×C₅
Hall	C₅×C₁₀₉ ⋊ C₂×D₁₇
Sylow	C₁₀₉ ⋊ C₂×C₅×D₁₇
Hall	C₅×C₁₇ ⋊ C₂×D₁₀₉
Sylow	C₁₇ ⋊ C₂×C₅×D₁₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₇×D₁₀₉
commutator, cosocle, hypocenter	C₁₇×C₁₀₉	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₇×C₁₀₉	C₂²
LCS	G ▷ C₁₇×C₁₀₉
UCS	C₁ ◁ C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 9265

A minimal faithful permutation representation — 2 generators on 131 points:

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

Conjugacy classes & character table

2800 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37060, 16);

C5×D17×D109 #49150385