Dic₇₇₇₇ #44527406

order	31108
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₇₇₇
unambiguous semidirect product	C₇⋊Dic₁₁₁₁
unambiguous semidirect product	C₁₁⋊Dic₇₀₇

Splittings

derived, Hall	C₇×C₁₁×C₁₀₁ ⋊ C₄
Hall	C₁₁×C₁₀₁ ⋊ Dic₇
Hall	C₇×C₁₀₁ ⋊ Dic₁₁
Sylow	C₁₀₁ ⋊ Dic₇₇
Hall	C₇×C₁₁ ⋊ Dic₁₀₁
Sylow	C₁₁ ⋊ Dic₇₀₇
Sylow	C₇ ⋊ Dic₁₁₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₇₇₇
commutator, hypocenter	C₇×C₁₁×C₁₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₁×C₁₀₁	C₂
LCS	G ▷ C₇×C₁₁×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 31108

A minimal faithful permutation representation — 2 generators on 123 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

2,7,11,101,2|a^2=e,b^a=b^6,c^a=c^10,d^a=d^100

Conjugacy classes & character table

7780 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₅₅₄
maximal normal subgroup of	Dic₁₅₅₅₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31108, 8);

Dic7777 #44527406