#27158508

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	AGL(1,ℤ/8)×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ #2955

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×#87604
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₄	C₂³×#87604
Fitting	AGL(1,ℤ/8)×C₉₇	C₄
socle	C₂×C₉₇	C₂×C₄×D₄
prime radical	#20261531	C₂
cosocle	C₄×D₉₇	C₂⁴
hypercenter	AGL(1,ℤ/8)	#87604
hypocenter	C₉₇	#2955
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

456 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:

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

Polycyclic presentation:

2^7,97|a^2=f,c^2=f,e^2=g,f^2=g,b^a=be,d^a=dg,e^a=eg,h^a=h^75,e^b=eg,h^c=h^22,h^f=h^96

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21282);