#27209917

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	#575×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #3584

Subgroups

	subgroup	G/H
center	C₄	C₂×D₄×D₁₀₃
commutator, cosocle	C₄×C₁₀₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₃
Fitting	#575×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂³×D₄
hypercenter	#575	D₁₀₃
hypocenter	C₁₀₃	#3584
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

1424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 3296

A minimal faithful permutation representation — 5 generators on 135 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)(104,108,110,109)(105,113,114,122)(106,117,118,123)(107,120,119,121)(111,125,126,115)(112,128,127,132)(116,131,130,133)(124,135,134,129);(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)(107,119)(112,127)(116,130)(120,121)(124,134)(128,132)(129,135)(131,133);(104,107)(105,112)(106,116)(108,120)(109,121)(110,119)(111,124)(113,128)(114,127)(115,129)(117,131)(118,130)(122,132)(123,133)(125,135)(126,134);(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)(104,106)(105,111)(107,116)(108,117)(109,123)(110,118)(112,124)(113,125)(114,126)(115,122)(119,130)(120,131)(121,133)(127,134)(128,135)(129,132);(104,105)(106,115)(107,112)(108,113)(109,122)(110,114)(111,117)(116,129)(118,125)(119,127)(120,128)(121,132)(123,126)(124,131)(130,135)(133,134)

Polycyclic presentation:

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

Conjugacy classes & character table

1166 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 19283);