Dic₇₈₃₁ #44531202

order	31324
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₈₃₁
unambiguous semidirect product	C₁₉₁⋊Dic₄₁

Splittings

derived, Hall	C₄₁×C₁₉₁ ⋊ C₄
Sylow	C₁₉₁ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₁₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₈₃₁
commutator, hypocenter	C₄₁×C₁₉₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₁₉₁	C₂
LCS	G ▷ C₄₁×C₁₉₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 236
transitive degree	μ_t(G) = 31324

A minimal faithful permutation representation — 2 generators on 236 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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236);(1,3,2,4)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(47,236)(48,235)(49,234)(50,233)(51,232)(52,231)(53,230)(54,229)(55,228)(56,227)(57,226)(58,225)(59,224)(60,223)(61,222)(62,221)(63,220)(64,219)(65,218)(66,217)(67,216)(68,215)(69,214)(70,213)(71,212)(72,211)(73,210)(74,209)(75,208)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,195)(89,194)(90,193)(91,192)(92,191)(93,190)(94,189)(95,188)(96,187)(97,186)(98,185)(99,184)(100,183)(101,182)(102,181)(103,180)(104,179)(105,178)(106,177)(107,176)(108,175)(109,174)(110,173)(111,172)(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,165)(119,164)(120,163)(121,162)(122,161)(123,160)(124,159)(125,158)(126,157)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)

Polycyclic presentation:

2,41,191,2|a^2=d,b^a=b^40,c^a=c^190

Conjugacy classes & character table

7834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₆₆₂
maximal normal subgroup of	Dic₁₅₆₆₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31324, 4);

Dic7831 #44531202