Dic₂₄₈ #11758389

order	992
automorphism	Aut(G) order 119040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₄₈

Splittings

Sylow

C₃₁ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₂₄₈
commutator, cosocle	C₈×C₃₁	C₂²
Frattini	C₈	C₂×D₃₁
Fitting	C₁₆×C₃₁	C₂
socle, prime radical	C₂×C₃₁	D₈
hypercenter	C₁₆	D₃₁
hypocenter	C₃₁	Q₃₂
LCS	G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 2 generators on 63 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);(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)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^5,31|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^30

Conjugacy classes & character table

251 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#14520887, #14520888, #14520889, …
maximal normal subgroup of	#14520887, #14520888, #14520889, …
maximal quotient of	#14520854, #14520875, C₂×Dic₂₄₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(992, 7);

Dic248 #11758389