Dic₆₇₅₅ #41028343

order	27020
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₇₅₅
unambiguous semidirect product	C₇⋊Dic₉₆₅

Splittings

derived, Hall	C₅×C₇×C₁₉₃ ⋊ C₄
Hall	C₇×C₁₉₃ ⋊ Dic₅
Hall	C₅×C₁₉₃ ⋊ Dic₇
Sylow	C₁₉₃ ⋊ Dic₃₅
Hall	C₅×C₇ ⋊ Dic₁₉₃
Sylow	C₇ ⋊ Dic₉₆₅
Sylow	C₅ ⋊ Dic₁₃₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₇₅₅
commutator, hypocenter	C₅×C₇×C₁₉₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₇×C₁₉₃	C₂
LCS	G ▷ C₅×C₇×C₁₉₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 209
transitive degree	μ_t(G) = 27020

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

Polycyclic presentation:

2,5,7,193,2|a^2=e,b^a=b^4,c^a=c^6,d^a=d^192

Conjugacy classes & character table

6758 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₅₁₀
maximal normal subgroup of	Dic₁₃₅₁₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27020, 8);

Dic6755 #41028343