Dic₂₃₂ #11745149

order	928
automorphism	Aut(G) order 103936
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₃₂

Splittings

Sylow

C₂₉ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₂₃₂
commutator, cosocle	C₈×C₂₉	C₂²
Frattini	C₈	C₂×D₂₉
Fitting	C₁₆×C₂₉	C₂
socle, prime radical	C₂×C₂₉	D₈
hypercenter	C₁₆	D₂₉
hypocenter	C₂₉	Q₃₂
LCS	G ▷ C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 2 generators on 61 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);(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)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^5,29|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^28

Conjugacy classes & character table

235 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#14271001, #14271002, #14271003, …
maximal normal subgroup of	#14271001, #14271002, #14271003, …
maximal quotient of	#14270968, #14270989, C₂×Dic₂₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(928, 8);

Dic232 #11745149