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C₂×C₉×B(4,𝔽₂) #11767744

order	1152
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂⁷

Aliases

C₂×C₉×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂³	C₂⁴×C₉
Frattini, cosocle	C₂³×C₃	C₂⁴×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂×C₃×B(4,𝔽₂)	C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

1527 conjugacy classes of subgroups; only the 324 normal subgroups are shown.

normal subgroups only

This group has 1527 conjugacy classes of subgroups — too many to draw. Only its 324 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) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2)(11,12,13,14,15,16,17,18,19);(7,10);(3,4)(5,7)(6,8)(9,10);(3,5)(4,7)(6,9)(8,10)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×D₉×B(4,𝔽₂), #72268842, #72268990
maximal subgroup of	B(4,𝔽₂)×AGL(1,𝔽₁₉), C₂×D₉×B(4,𝔽₂), #72268842, …
maximal normal subgroup of	C₂×D₉×B(4,𝔽₂), #72268842, C₉×#23900, …
maximal quotient of	B(4,𝔽₂)×AGL(1,𝔽₁₉), C₉×#23900, C₉×#32898
automorphism group of	C₂×D₄×C₁₉, C₂×D₄×C₂₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 1755);