#27156913

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄⋊C₄×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ C₄⋊#95

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₂×C₄⋊C₄×C₉₇	C₄
socle	C₂³×C₉₇	C₂²×C₄
prime radical, cosocle	C₂³×D₉₇	C₂³
hypercenter	C₂×C₄⋊C₄	#87604
hypocenter	C₉₇	C₄⋊#95
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 109 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

536 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 19687);