Dic₂₇₄₃ #24926622

order	10972
automorphism	Aut(G) order 13824720
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₄₃
unambiguous semidirect product	C₂₁₁⋊Dic₁₃

Splittings

derived, Hall	C₁₃×C₂₁₁ ⋊ C₄
Sylow	C₂₁₁ ⋊ Dic₁₃
Sylow	C₁₃ ⋊ Dic₂₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₇₄₃
commutator, hypocenter	C₁₃×C₂₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₂₁₁	C₂
LCS	G ▷ C₁₃×C₂₁₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 228
transitive degree	μ_t(G) = 10972

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

Polycyclic presentation:

2,13,211,2|a^2=d,b^a=b^12,c^a=c^210

Conjugacy classes & character table

2746 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₇₄₃, Dic₈₂₂₉, (C₂₁₁⋊C₃)⋊Dic₁₃, …
maximal normal subgroup of	C₃×Dic₂₇₄₃, (C₂₁₁⋊C₃)⋊Dic₁₃, #45681788, …
maximal quotient of	C₃×Dic₂₇₄₃, Dic₈₂₂₉, Dic₁₃₇₁₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10972, 4);

Dic2743 #24926622