Dic₃₀₈₈ #27135100

order	12352
automorphism	Aut(G) order 18972672
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₀₈₈

Splittings

Sylow

C₁₉₃ ⋊ Q₆₄

Subgroups

	subgroup	G/H
center	C₂	D₃₀₈₈
commutator, cosocle	C₁₆×C₁₉₃	C₂²
Frattini	C₁₆	C₂×D₁₉₃
Fitting	C₃₂×C₁₉₃	C₂
socle, prime radical	C₂×C₁₉₃	D₁₆
hypercenter	C₃₂	D₁₉₃
hypocenter	C₁₉₃	Q₆₄
LCS	G ▷ C₁₆×C₁₉₃ ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 257
transitive degree	μ_t(G) = 12352

A minimal faithful permutation representation — 2 generators on 257 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)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,257)(67,256)(68,255)(69,254)(70,253)(71,252)(72,251)(73,250)(74,249)(75,248)(76,247)(77,246)(78,245)(79,244)(80,243)(81,242)(82,241)(83,240)(84,239)(85,238)(86,237)(87,236)(88,235)(89,234)(90,233)(91,232)(92,231)(93,230)(94,229)(95,228)(96,227)(97,226)(98,225)(99,224)(100,223)(101,222)(102,221)(103,220)(104,219)(105,218)(106,217)(107,216)(108,215)(109,214)(110,213)(111,212)(112,211)(113,210)(114,209)(115,208)(116,207)(117,206)(118,205)(119,204)(120,203)(121,202)(122,201)(123,200)(124,199)(125,198)(126,197)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171)(153,170)(154,169)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)

Polycyclic presentation:

2^6,193|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^192,c^b=cd,d^b=de,e^b=ef,g^b=g^192

Conjugacy classes & character table

3091 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₂₆₄, Dic₁₅₄₄₀
maximal quotient of	Dic₉₂₆₄, Dic₁₅₄₄₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12352, 390);

Dic3088 #27135100