Dic₅₇₉₇ #37550847

order	23188
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₇₉₇
unambiguous semidirect product	C₁₁⋊Dic₅₂₇
unambiguous semidirect product	C₃₁⋊Dic₁₈₇

Splittings

derived, Hall	C₁₁×C₁₇×C₃₁ ⋊ C₄
Hall	C₁₇×C₃₁ ⋊ Dic₁₁
Hall	C₁₁×C₃₁ ⋊ Dic₁₇
Hall	C₁₁×C₁₇ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₈₇
Sylow	C₁₇ ⋊ Dic₃₄₁
Sylow	C₁₁ ⋊ Dic₅₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇₉₇
commutator, hypocenter	C₁₁×C₁₇×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₇×C₃₁	C₂
LCS	G ▷ C₁₁×C₁₇×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 23188

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)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(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:

2,11,17,31,2|a^2=e,b^a=b^10,c^a=c^16,d^a=d^30

Conjugacy classes & character table

5800 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₅₉₄
maximal normal subgroup of	Dic₁₁₅₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23188, 8);

Dic5797 #37550847