C₂×#4296⋊(C₇⋊D₇) #45692193

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

Aliases

direct product	C₂×#4296⋊(C₇⋊D₇)
unambiguous semidirect product	(C₂×#4296)⋊(C₇⋊D₇)

Splittings

Fitting, socle	C₂×C₇²×C₁₉ ⋊ C₂×C₉
derived, Hall	C₇²×C₁₉ ⋊ C₂²×C₉
Hall	C₇²×#4296 ⋊ C₂²
Hall	C₂×C₇⋊D₁₃₃ ⋊ C₉
Hall	#4296 ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×AGL(1,𝔽₁₉)
Sylow	C₁₉ ⋊ C₂×C₉×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#4296⋊(C₇⋊D₇)
commutator, hypocenter	C₇²×C₁₉	C₂²×C₉
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₉	C₂×C₉
prime radical	C₂×(C₁₉⋊C₃)⋊(C₇⋊D₇)	C₃
cosocle	C₇²×C₁₉⋊C₃	C₂²×C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₂

300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 1862

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 991);

C2×#4296⋊(C7⋊D7) #45692193