#27158175

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×D₄×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ #2275

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87604
commutator	C₂×C₉₇	C₂²×C₄²
Fitting	C₄×D₄×C₉₇	C₄
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂×D₄×D₉₇	C₂²
cosocle	C₂²×D₉₇	C₂⁴
hypercenter	C₄×D₄	#87604
hypocenter	C₉₇	#2275
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 4 generators on 109 points:

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

Polycyclic presentation:

2^7,97|a^2=f,b^2=g,c^2=f,b^a=be,d^a=de,h^a=h^75,c^b=ce,h^c=h^22,h^f=h^96

Conjugacy classes & character table

548 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 20949);