Dic₄₂₄ #13133164

order	1696
automorphism	Aut(G) order 352768
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₂₄

Splittings

Sylow

C₅₃ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₄₂₄
commutator, cosocle	C₈×C₅₃	C₂²
Frattini	C₈	C₂×D₅₃
Fitting	C₁₆×C₅₃	C₂
socle, prime radical	C₂×C₅₃	D₈
hypercenter	C₁₆	D₅₃
hypocenter	C₅₃	Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 85 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^5,53|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^52,c^b=cd,d^b=de,f^b=f^52

Conjugacy classes & character table

427 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#16810461, #16810462, #16810463, …
maximal normal subgroup of	#16810461, #16810462, #16810463, …
maximal quotient of	#16810428, #16810449, C₂×Dic₄₂₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1696, 93);

Dic424 #13133164