Dic₃₀₀₇ #26046689

order	12028
automorphism	Aut(G) order 17320320
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₀₀₇
unambiguous semidirect product	C₃₁⋊Dic₉₇

Splittings

derived, Hall	C₃₁×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₀₀₇
commutator, hypocenter	C₃₁×C₉₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₁×C₉₇	C₂
LCS	G ▷ C₃₁×C₉₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 12028

A minimal faithful permutation representation — 2 generators on 132 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);(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)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)

Polycyclic presentation:

2,31,97,2|a^2=d,b^a=b^30,c^a=c^96

Conjugacy classes & character table

3010 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₃₀₀₇, Dic₉₀₂₁, #49115323, …
maximal normal subgroup of	C₃×Dic₃₀₀₇, #49115323, (C₃₁⋊C₃)⋊Dic₉₇, …
maximal quotient of	C₃×Dic₃₀₀₇, Dic₉₀₂₁, Dic₁₅₀₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12028, 4);

Dic3007 #26046689