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C₂⁴×D₁₂ #87554

order	384
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

C₂⁴×D₁₂

Splittings

Fitting	C₂⁴×C₃×C₄ ⋊ C₂
Sylow	C₃ ⋊ C₂⁴×D₄

Subgroups

	subgroup	G/H
center	C₂⁵	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂⁴×C₃×C₄	C₂
socle	C₂⁵×C₃	C₂²
hypercenter	C₂⁴×C₄	D₃
hypocenter	C₃	C₂⁴×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

10552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 6 generators on 15 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#69426, #10717056, #10717402, …
maximal normal subgroup of	C₂×#69426, #10717056, #10717402, …
maximal quotient of	C₂⁴×(C₂×D₃)⋊C₄, C₂⁴×C₄⋊Dic₃, C₂³×C₄×D₁₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20130);