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C₂²×D₃×C₁₆ #81998

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×D₃×C₁₆

Splittings

Fitting	C₂²×C₃×C₁₆ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂³×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	D₃
commutator, hypocenter	C₃	C₂³×C₁₆
Frattini	C₈	C₂³×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle	C₂³×C₃	C₂×C₈
prime radical	C₂³×D₃	C₈
cosocle	C₃×C₈	C₂⁴
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₁₆

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1374, #10710028, D₃×#1379, …
maximal normal subgroup of	D₃×#1374, #10710028, D₃×#1379, …
maximal quotient of	C₂×C₄×D₃×C₁₆, C₂²×Dic₃×C₁₆, C₂×D₃×C₂²⋊C₁₆, …
automorphism group of	D₃×C₆₄, #67432, C₁₇×C₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14574);