D₃×C₇×D₃₂₁ #41004134

order	26964
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×C₇×D₃₂₁
unambiguous semidirect product	(C₃×D₃)⋊(C₇×D₁₀₇)
unambiguous semidirect product	(D₃×C₁₀₇)⋊(D₃×C₇)

Splittings

Fitting, socle, Hall	C₃²×C₇×C₁₀₇ ⋊ C₂²
derived, Hall	C₃²×C₁₀₇ ⋊ C₂²×C₇
Hall	C₇×C₁₀₇ ⋊ D₃²
Sylow	C₁₀₇ ⋊ D₃²×C₇
Hall	C₃²×C₇ ⋊ C₂×D₁₀₇
Sylow	C₃² ⋊ C₂×C₇×D₁₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₃×D₃₂₁
commutator, cosocle, hypocenter	C₃²×C₁₀₇	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₁₀₇	C₂²
LCS	G ▷ C₃²×C₁₀₇
UCS	C₁ ◁ C₇

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 4494

A minimal faithful permutation representation — 2 generators on 120 points:

(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)(112,113)(114,115,116,117,118,119,120);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(109,110)(111,112,113)

Conjugacy classes & character table

3402 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26964, 83);

D3×C7×D321 #41004134