Dic₆₁₀₇ #37567682

order	24428
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₁₀₇
unambiguous semidirect product	C₃₁⋊Dic₁₉₇

Splittings

derived, Hall	C₃₁×C₁₉₇ ⋊ C₄
Sylow	C₁₉₇ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₁₀₇
commutator, hypocenter	C₃₁×C₁₉₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₁×C₁₉₇	C₂
LCS	G ▷ C₃₁×C₁₉₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 232
transitive degree	μ_t(G) = 24428

A minimal faithful permutation representation — 2 generators on 232 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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232);(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)(37,232)(38,231)(39,230)(40,229)(41,228)(42,227)(43,226)(44,225)(45,224)(46,223)(47,222)(48,221)(49,220)(50,219)(51,218)(52,217)(53,216)(54,215)(55,214)(56,213)(57,212)(58,211)(59,210)(60,209)(61,208)(62,207)(63,206)(64,205)(65,204)(66,203)(67,202)(68,201)(69,200)(70,199)(71,198)(72,197)(73,196)(74,195)(75,194)(76,193)(77,192)(78,191)(79,190)(80,189)(81,188)(82,187)(83,186)(84,185)(85,184)(86,183)(87,182)(88,181)(89,180)(90,179)(91,178)(92,177)(93,176)(94,175)(95,174)(96,173)(97,172)(98,171)(99,170)(100,169)(101,168)(102,167)(103,166)(104,165)(105,164)(106,163)(107,162)(108,161)(109,160)(110,159)(111,158)(112,157)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)(119,150)(120,149)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)

Polycyclic presentation:

2,31,197,2|a^2=d,b^a=b^30,c^a=c^196

Conjugacy classes & character table

6110 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₂₁₄
maximal normal subgroup of	Dic₁₂₂₁₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24428, 4);

Dic6107 #37567682