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#41×Dic₆ #85824

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

Splittings

C₃ ⋊ Q₈×#41

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₃×C₄×#41	C₂
prime radical	C₂×C₃×#41	C₂²
hypercenter	C₄×#41	D₃
hypocenter	C₃	Q₈×#41
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 19 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10687229, #10687233, #10762062, …
maximal normal subgroup of	#10687229, #10687233, #10762062, …
maximal quotient of	#41×Dic₃⋊C₄, #41×C₄⋊Dic₃, Dic₆×#117, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18400);