Dic₆₃₂₈ #72236064

order	25312
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₃₂₈

Splittings

Hall	C₇×C₁₁₃ ⋊ Q₃₂
Sylow	C₁₁₃ ⋊ Dic₅₆
Sylow	C₇ ⋊ Dic₉₀₄

Subgroups

	subgroup	G/H
center	C₂	D₆₃₂₈
commutator, cosocle	C₇×C₈×C₁₁₃	C₂²
Frattini	C₈	C₂×D₇₉₁
Fitting	C₇×C₁₆×C₁₁₃	C₂
socle, prime radical	C₂×C₇×C₁₁₃	D₈
hypercenter	C₁₆	D₇₉₁
hypocenter	C₇×C₁₁₃	Q₃₂
LCS	G ▷ C₇×C₈×C₁₁₃ ▷ C₄×C₇×C₁₁₃ ▷ C₂×C₇×C₁₁₃ ▷ C₇×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 152
transitive degree	μ_t(G) = 25312

A minimal faithful permutation representation — 2 generators on 152 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,39)(35,38)(36,37)(41,152)(42,151)(43,150)(44,149)(45,148)(46,147)(47,146)(48,145)(49,144)(50,143)(51,142)(52,141)(53,140)(54,139)(55,138)(56,137)(57,136)(58,135)(59,134)(60,133)(61,132)(62,131)(63,130)(64,129)(65,128)(66,127)(67,126)(68,125)(69,124)(70,123)(71,122)(72,121)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)

Polycyclic presentation:

2^5,7,113|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^112

Conjugacy classes & character table

6331 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₆₅₆
maximal normal subgroup of	Dic₁₂₆₅₆

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 7, 113]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6328 #72236064