D₃×AGL(1,𝔽₃₇) #72228577

order	7992
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

D₃×AGL(1,𝔽₃₇)

Splittings

Fitting, socle, derived	C₃×C₃₇ ⋊ C₂×C₄×C₉
Hall	C₃×#66526 ⋊ C₂×C₄
Sylow	C₃₇ ⋊ C₄×D₃×C₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₃₇	C₂×C₄×C₉
prime radical	D₃×D₃₇⋊C₃	C₂×C₃
cosocle	C₃×D₃₇⋊C₃	C₂²×C₃
LCS	G ▷ C₃×C₃₇
UCS	C₁

108 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 108 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 111

A minimal faithful permutation representation — 2 generators on 40 points:

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

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×AGL(1,𝔽₃₇), C₂×D₃×AGL(1,𝔽₃₇), C₃⋊D₃×AGL(1,𝔽₃₇)
maximal normal subgroup of	C₂×D₃×AGL(1,𝔽₃₇)
maximal quotient of	S₄×AGL(1,𝔽₃₇), C₂×D₃×AGL(1,𝔽₃₇), C₃⋊D₃×AGL(1,𝔽₃₇)
automorphism group of	D₁₁₁
holomorph of	C₃×C₃₇

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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