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C₂²×C₄×C₁₉⋊C₃ #11744952

order	912
isoclinism	isoclinic class of C₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₄×C₁₉⋊C₃

Splittings

Fitting, Hall	C₂²×C₄×C₁₉ ⋊ C₃
derived, Sylow	C₁₉ ⋊ C₂²×C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	C₁₉⋊C₃
commutator, hypocenter	C₁₉	C₂²×C₃×C₄
Frattini	C₂	C₂³×C₁₉⋊C₃
Fitting	C₂²×C₄×C₁₉	C₃
socle	C₂³×C₁₉	C₂×C₃
prime radical	C₂³×C₁₉⋊C₃	C₂
cosocle	C₂×C₁₉	C₂³×C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 304

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂²⋊(C₁₉⋊C₈))⋊C₃, (C₁₉⋊C₃)⋊#95, #95×C₁₉⋊C₃, …
maximal normal subgroup of	(C₂²⋊(C₁₉⋊C₈))⋊C₃, (C₁₉⋊C₃)⋊#95, #95×C₁₉⋊C₃, …
maximal quotient of	C₂×C₄²×C₁₉⋊C₃, C₂×C₂²⋊C₄×C₁₉⋊C₃, C₂×C₄⋊C₄×C₁₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(912, 163);