Dic₅₃₆₃ #36339797

order	21452
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₃₆₃
unambiguous semidirect product	C₃₁⋊Dic₁₇₃

Splittings

derived, Hall	C₃₁×C₁₇₃ ⋊ C₄
Sylow	C₁₇₃ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₃₆₃
commutator, hypocenter	C₃₁×C₁₇₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₁×C₁₇₃	C₂
LCS	G ▷ C₃₁×C₁₇₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 208
transitive degree	μ_t(G) = 21452

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

Polycyclic presentation:

2,31,173,2|a^2=d,b^a=b^30,c^a=c^172

Conjugacy classes & character table

5366 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₇₂₆, Dic₁₆₀₈₉
maximal normal subgroup of	Dic₁₀₇₂₆
maximal quotient of	Dic₁₆₀₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21452, 4);

Dic5363 #36339797