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C₂²×D₄₈ #82001

order	384
isoclinism	isoclinic class of D₄₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×D₄₈

Splittings

Fitting	C₂²×C₃×C₁₆ ⋊ C₂
Sylow	C₃ ⋊ C₂²×D₁₆

Subgroups

	subgroup	G/H
center	C₂³	D₂₄
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle	C₂³×C₃	D₈
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	C₂²×D₁₆
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ 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:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10712243, #10712245, #10712278, …
maximal normal subgroup of	#10712243, #10712245, #10712278, …
maximal quotient of	C₂²×#4509, C₂²×#4504, C₂×C₄×D₄₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14577);