Dic₅₆₁₁ #36376277

order	22444
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₆₁₁
unambiguous semidirect product	C₃₁⋊Dic₁₈₁

Splittings

derived, Hall	C₃₁×C₁₈₁ ⋊ C₄
Sylow	C₁₈₁ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₆₁₁
commutator, hypocenter	C₃₁×C₁₈₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₁×C₁₈₁	C₂
LCS	G ▷ C₃₁×C₁₈₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 216
transitive degree	μ_t(G) = 22444

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

Polycyclic presentation:

2,31,181,2|a^2=d,b^a=b^30,c^a=c^180

Conjugacy classes & character table

5614 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₂₂₂
maximal normal subgroup of	Dic₁₁₂₂₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22444, 4);

Dic5611 #36376277