#27158925

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #2150

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715906
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	C₄²×C₉₇	C₈
prime radical	C₂²×D₉₇	C₂×C₈
cosocle	C₂×#11715897	C₂³
hypercenter	C₄²	#11715906
hypocenter	C₉₇	#2150
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄²

172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 117 points:

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

Polycyclic presentation:

2^7,97|a^2=e,b^2=d,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^64,h^c=h^96,h^e=h^22,h^f=h^96

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21699);