#27197031

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	#346×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #2179

Subgroups

	subgroup	G/H
center	C₄	C₂×D₈₂₄
commutator	C₈×C₁₀₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₃
Fitting	#346×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₄×D₈
prime radical	#20268223	C₂
cosocle	C₂×C₈×C₁₀₃	C₂³
hypercenter	#346	D₁₀₃
hypocenter	C₁₀₃	#2179
LCS	G ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(104,107,109,122,110,123,126,108)(105,112,114,124,116,113,115,131)(106,118,119,127,121,135,120,125)(111,128,129,117,130,132,133,134);(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)(107,118)(108,125)(109,119)(110,121)(112,128)(113,132)(114,129)(115,133)(116,130)(117,124)(120,126)(122,127)(123,135)(131,134);(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)(104,105,109,114,110,116,126,115)(106,117,119,132,121,134,120,128)(107,113,122,131,123,112,108,124)(111,127,129,135,130,125,133,118)

Polycyclic presentation:

2^7,103|a^2=e,c^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^102,d^b=df,f^b=fg,h^b=h^102

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 6397);