C₅×ΓL(1,𝔽₈)×C₁₉₁ #34095381

order	20055
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃×C₅×C₁₉₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×ΓL(1,𝔽₈)×C₁₉₁
unambiguous semidirect product	C₇⋊(C₃×C₅×C₁₉₁)
unambiguous semidirect product	(C₅×C₇)⋊(C₃×C₁₉₁)
unambiguous semidirect product	(C₅×C₇×C₁₉₁)⋊C₃

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₉₁ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃×C₅×C₁₉₁
Hall	C₇×C₁₉₁ ⋊ C₃×C₅
Hall	C₅×C₇ ⋊ C₃×C₁₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₁₉₁	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₅×C₁₉₁
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₉₁	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₅×C₁₉₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 203
transitive degree	μ_t(G) = 6685

A minimal faithful permutation representation — 2 generators on 203 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);(1,2,3,5,4,6,7)

Conjugacy classes & character table

4775 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20055, 3);

C5×ΓL(1,𝔽8)×C191 #34095381