#27158991

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #3407

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×#11715906
commutator, socle	C₂×C₉₇	C₂³×C₈
Fitting	C₂×Q₈×C₉₇	C₈
prime radical	C₂×#13095512	C₄
cosocle	#11715897	C₂⁴
hypercenter	C₂×Q₈	#11715906
hypocenter	C₉₇	#3407
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 129 points:

(1,2,6,15,31,58,82,93,79,61,68,40,66,86,46,74,71,24,50,72,54,26,5,11,27,43,70,87,59,84,65,12,20,7,17,36,63,88,96,67,92,89,33,52,14,29,37,16,32,41,8,18,39,28,53,38,64,90,91,97,95,81,55,34,3,9,22,45,73,35,60,69,21,44,25,10,23,47,75,78,48,51,80,49,76,77,19,42,57,83,94,62,85,56,30,13,4)(99,105)(104,114)(106,117)(107,118)(115,124)(116,125)(119,127)(126,129);(2,4)(3,7)(5,10)(6,13)(8,16)(9,20)(11,25)(12,22)(14,28)(15,30)(17,34)(18,37)(19,40)(21,43)(23,26)(24,48)(27,44)(29,39)(31,56)(32,41)(33,38)(35,59)(36,55)(42,68)(45,65)(46,49)(47,54)(50,78)(51,71)(52,53)(57,61)(58,85)(60,87)(62,82)(63,81)(64,89)(66,77)(67,91)(69,70)(72,75)(73,84)(74,80)(76,86)(79,83)(88,95)(90,92)(93,94)(96,97)(100,108)(104,114)(109,120)(110,121)(115,124)(116,125)(122,128)(126,129);(2,4)(3,7)(5,10)(6,13)(8,16)(9,20)(11,25)(12,22)(14,28)(15,30)(17,34)(18,37)(19,40)(21,43)(23,26)(24,48)(27,44)(29,39)(31,56)(32,41)(33,38)(35,59)(36,55)(42,68)(45,65)(46,49)(47,54)(50,78)(51,71)(52,53)(57,61)(58,85)(60,87)(62,82)(63,81)(64,89)(66,77)(67,91)(69,70)(72,75)(73,84)(74,80)(76,86)(79,83)(88,95)(90,92)(93,94)(96,97)(98,100)(99,104)(101,109)(102,110)(103,108)(105,114)(106,115)(107,116)(111,122)(112,120)(113,121)(117,124)(118,125)(119,126)(123,128)(127,129);(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)(98,99,101,106,102,107,111,119,103,105,112,117,113,118,123,127)(100,104,109,115,110,116,122,126,108,114,120,124,121,125,128,129)

Polycyclic presentation:

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

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21765);