C₇⋊D₇×AGL(1,𝔽₁₉) #45692097

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

Aliases

direct product	C₇⋊D₇×AGL(1,𝔽₁₉)
unambiguous semidirect product	(C₇×AGL(1,𝔽₁₉))⋊D₇

Splittings

Fitting, socle, derived, Hall	C₇²×C₁₉ ⋊ C₂²×C₉
Hall	C₇²×#4296 ⋊ C₂²
Hall	D₁₉×C₇⋊D₇ ⋊ C₉
Hall	#4296 ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×AGL(1,𝔽₁₉)
Sylow	C₁₉ ⋊ C₂×C₉×C₇⋊D₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₁₉	C₂²×C₉
prime radical	C₇⋊D₇×D₁₉⋊C₃	C₃
cosocle	C₇²×C₁₉⋊C₃	C₂²×C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁

300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 931

A minimal faithful permutation representation — 3 generators on 33 points:

(8,9,11,13,14,12,10)(16,27,28,21,30,26,24,20,29,17,23,33,19,31,22,25,18,32);(2,3)(4,5)(6,7)(9,10)(11,12)(13,14)(15,16,20,27,22,30,29,24,21,28,33,19,25,32,31,26,23,18,17);(1,2,4,6,7,5,3)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 895);

C7⋊D7×AGL(1,𝔽19) #45692097