Dic₄₄₀₃ #31798548

order	17612
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₄₀₃
unambiguous semidirect product	C₇⋊Dic₆₂₉

Splittings

derived, Hall	C₇×C₁₇×C₃₇ ⋊ C₄
Hall	C₁₇×C₃₇ ⋊ Dic₇
Hall	C₇×C₃₇ ⋊ Dic₁₇
Hall	C₇×C₁₇ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₁₁₉
Sylow	C₁₇ ⋊ Dic₂₅₉
Sylow	C₇ ⋊ Dic₆₂₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₄₀₃
commutator, hypocenter	C₇×C₁₇×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₃₇	C₂
LCS	G ▷ C₇×C₁₇×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 17612

A minimal faithful permutation representation — 2 generators on 65 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(1,3,2,4)(6,11)(7,10)(8,9)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,65)(31,64)(32,63)(33,62)(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:

2,7,17,37,2|a^2=e,b^a=b^6,c^a=c^16,d^a=d^36

Conjugacy classes & character table

4406 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₈₀₆, Dic₁₃₂₀₉
maximal normal subgroup of	Dic₈₈₀₆
maximal quotient of	Dic₁₃₂₀₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17612, 8);

Dic4403 #31798548