#27214005

order	13509
automorphism	?
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived, Hall	C₁₉×C₇₉ ⋊ C₉
Sylow	C₇₉ ⋊ #4294
Sylow	C₁₉ ⋊ C₇₉⋊C₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	#17985163
commutator, hypocenter	C₁₉×C₇₉	C₉
Fitting, socle, prime radical, cosocle	C₃×C₁₉×C₇₉	C₃
LCS	G ▷ C₁₉×C₇₉
UCS	C₁ ◁ C₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 13509

A minimal faithful permutation representation — 2 generators on 107 points:

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

Polycyclic presentation:

3,19,79,3|a^3=d,b^a=b^7,c^a=c^23

Conjugacy classes & character table

1509 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#27214005, #41028265, #41028269, …
maximal normal subgroup of	C₂×#27214005, #41028265, #41028269, …
maximal quotient of	C₂×#27214005

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13509, 4);