#27193354

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₆₄×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ M₁₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₃₂	C₂×D₁₀₃
commutator, socle	C₂×C₁₀₃	C₂×C₃₂
Fitting	C₆₄×C₁₀₃	C₂
prime radical	C₂×D₁₀₃	C₃₂
cosocle	C₃₂×C₁₀₃	C₂²
hypercenter	C₆₄	D₁₀₃
hypocenter	C₁₀₃	M₁₂₈
LCS	G ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₃₂ ◁ C₆₄

38 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)(112,126)(113,127)(114,128)(115,129)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(152,161)(153,162)(154,163)(155,164)(165,167);(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)(104,105,106,112,107,113,116,130,108,114,117,131,120,133,136,152,109,115,118,132,121,134,137,153,123,135,139,154,142,155,156,165,110,111,119,126,122,127,138,146,124,128,140,147,143,149,157,161,125,129,141,148,144,150,158,162,145,151,159,163,160,164,166,167)

Polycyclic presentation:

2^7,103|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,b^a=bg,h^b=h^102

Conjugacy classes & character table

3344 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 2720);