C₉×Dic₁₂₁ #17983676

order	4356
isoclinism	isoclinic class of D₁₂₁
automorphism	?
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×Dic₁₂₁
unambiguous semidirect product	C₁₂₁⋊(C₄×C₉)

Splittings

derived, Sylow	C₁₂₁ ⋊ C₄×C₉
Hall	C₉×C₁₂₁ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₉	D₁₂₁
commutator, hypocenter	C₁₂₁	C₄×C₉
Frattini, socle, prime radical	C₂×C₃×C₁₁	C₃×D₁₁
Fitting	C₂×C₉×C₁₂₁	C₂
cosocle	C₂×C₃×C₁₂₁	C₂×C₃
LCS	G ▷ C₁₂₁
UCS	C₁ ◁ C₂×C₉

27 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 4356

A minimal faithful permutation representation — 2 generators on 134 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);(10,12,11,13)(15,134)(16,133)(17,132)(18,131)(19,130)(20,129)(21,128)(22,127)(23,126)(24,125)(25,124)(26,123)(27,122)(28,121)(29,120)(30,119)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Conjugacy classes & character table

1116 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×Dic₁₂₁⋊C₅, C₅×C₉×Dic₁₂₁, C₉×Dic₆₀₅, …
maximal normal subgroup of	C₉×Dic₁₂₁⋊C₅, C₅×C₉×Dic₁₂₁, C₇×C₉×Dic₁₂₁
maximal quotient of	C₅×C₉×Dic₁₂₁, C₉×Dic₆₀₅, C₉×C₁₂₁⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4356, 3);

C9×Dic121 #17983676