#27158787

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	Q₁₆×C₉₇ ⋊ C₈
Sylow	C₉₇ ⋊ #1335

Subgroups

	subgroup	G/H
center	C₂	D₄⋊#11715906
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₈	C₂×#11715906
Fitting	Q₁₆×C₉₇	C₈
socle	C₂×C₉₇	D₄⋊C₈
prime radical	#15690833	C₄
cosocle	C₈×#87604	C₂²
hypercenter	Q₁₆	#11715906
hypocenter	C₉₇	#1335
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 2 generators on 113 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)(98,100)(99,104)(101,110)(102,107)(103,111)(105,113)(106,108)(109,112);(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)(98,99,101,106,103,107,113,109,104,100,108,110,102,111,112,105)

Polycyclic presentation:

2^7,97|a^2=d,c^2=eg,d^2=f,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,h^a=h^64,c^b=ce,e^b=eg,h^b=h^96,h^d=h^22,h^f=h^96

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21561);