D₁₅₆₁₇ #44530311

order	31234
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₆₁₇
unambiguous semidirect product	C₇⋊D₂₂₃₁
unambiguous semidirect product	C₂₃⋊D₆₇₉
unambiguous semidirect product	C₉₇⋊D₁₆₁

Splittings

Fitting, socle, derived, Hall	C₇×C₂₃×C₉₇ ⋊ C₂
Hall	C₂₃×C₉₇ ⋊ D₇
Hall	C₇×C₉₇ ⋊ D₂₃
Hall	C₇×C₂₃ ⋊ D₉₇
Sylow	C₉₇ ⋊ D₁₆₁
Sylow	C₂₃ ⋊ D₆₇₉
Sylow	C₇ ⋊ D₂₂₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₃×C₉₇	C₂
LCS	G ▷ C₇×C₂₃×C₉₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 15617

A minimal faithful permutation representation — 2 generators on 127 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);(2,7)(3,6)(4,5)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)

Polycyclic presentation:

2,7,23,97|b^a=b^6,c^a=c^22,d^a=d^96

Conjugacy classes & character table

7810 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₆₁₇
maximal normal subgroup of	C₂×D₁₅₆₁₇
maximal quotient of	Dic₁₅₆₁₇, C₂×D₁₅₆₁₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31234, 8);

D15617 #44530311