C₃×ΓL(1,𝔽₈)×C₂₁₁ #27210820

order	13293
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₂₁₁ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×ΓL(1,𝔽₈)×C₂₁₁
unambiguous semidirect product	C₇⋊(C₃²×C₂₁₁)
unambiguous semidirect product	(C₃×C₇)⋊(C₃×C₂₁₁)

Splittings

Fitting, socle	C₃×C₇×C₂₁₁ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃²×C₂₁₁
Hall	C₇×C₂₁₁ ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₂₁₁	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃²×C₂₁₁
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₂₁₁	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₂₁₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 221
transitive degree	μ_t(G) = 4431

A minimal faithful permutation representation — 2 generators on 221 points:

(2,3,4)(5,7,6)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218);(1,2,3,5,4,6,7)(219,220,221)

Conjugacy classes & character table

3165 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×AGL(1,𝔽₇)×C₂₁₁, C₃×ΓL(1,𝔽₈)⋊D₂₁₁, C₃×ΓL(1,𝔽₈)×D₂₁₁, …
maximal normal subgroup of	C₃×AGL(1,𝔽₇)×C₂₁₁, C₃×ΓL(1,𝔽₈)⋊D₂₁₁, C₃×ΓL(1,𝔽₈)×D₂₁₁, …
maximal quotient of	C₂×C₃×ΓL(1,𝔽₈)×C₂₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13293, 14);

C3×ΓL(1,𝔽8)×C211 #27210820