C₂×C₉×C₇⋊D₁₃₃ #45691713

order	33516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₇

Aliases

direct product

C₂×C₉×C₇⋊D₁₃₃

Splittings

Fitting	C₂×C₇²×C₉×C₁₉ ⋊ C₂
derived, Hall	C₇²×C₁₉ ⋊ C₂²×C₉
Hall	C₇²×C₉×C₁₉ ⋊ C₂²
Hall	C₇²×C₉ ⋊ C₂×D₁₉
Hall	C₉×C₁₉ ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×C₉×D₁₉
Sylow	C₁₉ ⋊ C₂×C₉×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₉	C₇⋊D₁₃₃
commutator, hypocenter	C₇²×C₁₉	C₂²×C₉
Frattini	C₃	C₂×C₃×C₇⋊D₁₃₃
Fitting	C₂×C₇²×C₉×C₁₉	C₂
socle	C₂×C₃×C₇²×C₁₉	C₂×C₃
prime radical	C₂×C₃×C₇⋊D₁₃₃	C₃
cosocle	C₃×C₇²×C₁₉	C₂²×C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₂×C₉

300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 16758

A minimal faithful permutation representation — 3 generators on 44 points:

(4,5)(6,7)(8,9)(11,12)(13,14)(15,16)(17,21,25,19,24,20,22,18,23)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44);(1,2)(10,11,13,15,16,14,12)(26,27,29,31,33,35,37,39,41,43,44,42,40,38,36,34,32,30,28);(3,4,6,8,9,7,5)

Conjugacy classes & character table

8406 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 511);

C2×C9×C7⋊D133 #45691713