Dic₇₄₃₉ #44446646

order	29756
automorphism	Aut(G) order 107478672
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₄₃₉
unambiguous semidirect product	C₄₃⋊Dic₁₇₃

Splittings

derived, Hall	C₄₃×C₁₇₃ ⋊ C₄
Sylow	C₁₇₃ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₄₃₉
commutator, hypocenter	C₄₃×C₁₇₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₃×C₁₇₃	C₂
LCS	G ▷ C₄₃×C₁₇₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 220
transitive degree	μ_t(G) = 29756

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

Polycyclic presentation:

2,43,173,2|a^2=d,b^a=b^42,c^a=c^172

Conjugacy classes & character table

7442 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₈₇₈
maximal normal subgroup of	Dic₁₄₈₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29756, 6);

Dic7439 #44446646