#27204362

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#41×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #3002

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₃
commutator, cosocle	C₂×C₄×C₁₀₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	C₂²×#41×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
hypercenter	C₂²×#41	D₁₀₃
hypocenter	C₁₀₃	#3002
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 3296

A minimal faithful permutation representation — 4 generators on 123 points:

(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)(108,111,113,112)(109,115,116,120)(110,118,119,121)(114,122,123,117);(105,107);(108,110)(109,114)(111,118)(112,121)(113,119)(115,122)(116,123)(117,120);(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)(104,105)(106,107)(108,109)(110,117)(111,115)(112,120)(113,116)(114,118)(119,122)(121,123)

Polycyclic presentation:

2^7,103|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^102,e^b=eg

Conjugacy classes & character table

2078 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 13728);