D₁₁₉₂₁ #37558646

order	23842
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉₂₁
unambiguous semidirect product	C₇⋊D₁₇₀₃
unambiguous semidirect product	C₁₃⋊D₉₁₇
unambiguous semidirect product	C₁₃₁⋊D₉₁

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₁₃₁ ⋊ C₂
Hall	C₁₃×C₁₃₁ ⋊ D₇
Hall	C₇×C₁₃₁ ⋊ D₁₃
Sylow	C₁₃₁ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₁₃₁
Sylow	C₁₃ ⋊ D₉₁₇
Sylow	C₇ ⋊ D₁₇₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₁₃₁	C₂
LCS	G ▷ C₇×C₁₃×C₁₃₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 151
transitive degree	μ_t(G) = 11921

A minimal faithful permutation representation — 2 generators on 151 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151);(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2,7,13,131|b^a=b^6,c^a=c^12,d^a=d^130

Conjugacy classes & character table

5962 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23842, 9);

D11921 #37558646