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D₄×ΓL(1,𝔽₂₅) #79424

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

D₄×ΓL(1,𝔽₂₅)

Splittings

Fitting	C₃×C₈×D₄ ⋊ C₂
Sylow	C₃ ⋊ D₄×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₈×D₄	C₂
prime radical	C₂×D₃×D₄	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₈×D₄	D₃
hypocenter	C₃	D₄×M₁₆
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10766068, #10766074, #10766079, …
maximal normal subgroup of	#10766068, #10766074, #10766079, …
maximal quotient of	D₈×ΓL(1,𝔽₂₅), #10766612, SD₁₆×ΓL(1,𝔽₂₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12000);