#27193356

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	D₃₂×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ D₆₄

Subgroups

	subgroup	G/H
center	C₂	#20267626
commutator, cosocle	C₃₂×C₁₀₃	C₂²
Frattini	C₃₂	C₂×D₁₀₃
Fitting	D₃₂×C₁₀₃	C₂
socle	C₂×C₁₀₃	D₃₂
hypercenter	D₃₂	D₁₀₃
hypocenter	C₁₀₃	D₆₄
LCS	G ▷ C₃₂×C₁₀₃ ▷ C₁₆×C₁₀₃ ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ D₃₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 167
transitive degree	μ_t(G) = 6592

A minimal faithful permutation representation — 2 generators on 167 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(105,111)(106,117)(107,118)(108,123)(109,127)(112,113)(114,139)(115,141)(116,133)(119,149)(120,152)(121,145)(122,130)(124,158)(125,148)(126,131)(128,156)(129,132)(134,154)(135,153)(136,138)(137,160)(140,164)(142,143)(144,150)(146,147)(151,167)(155,162)(157,161)(159,163)(165,166);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(104,105)(106,111)(107,122)(108,126)(109,129)(110,116)(112,117)(113,118)(114,123)(115,127)(119,139)(120,141)(121,133)(124,157)(125,155)(128,159)(130,143)(131,142)(132,150)(134,149)(135,152)(136,145)(137,158)(138,148)(140,156)(144,160)(146,154)(147,153)(151,164)(161,167)(162,166)(163,165)

Polycyclic presentation:

2^7,103|c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^102

Conjugacy classes & character table

1004 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 2722);