Dic₃₃₂₅ #27210850

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₃₂₅
unambiguous semidirect product	C₇⋊Dic₄₇₅
unambiguous semidirect product	C₁₉⋊Dic₁₇₅

Splittings

derived, Hall	C₇×C₁₉×C₂₅ ⋊ C₄
Hall	C₁₉×C₂₅ ⋊ Dic₇
Hall	C₇×C₂₅ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ Dic₁₃₃
Sylow	C₁₉ ⋊ Dic₁₇₅
Sylow	C₇ ⋊ Dic₄₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₃₂₅
commutator, hypocenter	C₇×C₁₉×C₂₅	C₄
Frattini	C₂×C₅	D₆₆₅
Fitting, cosocle	C₂×C₇×C₁₉×C₂₅	C₂
socle, prime radical	C₂×C₅×C₇×C₁₉	D₅
LCS	G ▷ C₇×C₁₉×C₂₅
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 13300

A minimal faithful permutation representation — 2 generators on 55 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);(1,3,2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,36)(32,35)(33,34)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,5,7,19,2,5|a^2=e,b^5=f^4,b^a=b^4f,c^a=c^6,d^a=d^18,f^a=f^4

Conjugacy classes & character table

3328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₆₅₀
maximal normal subgroup of	Dic₆₆₅₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 8);

Dic3325 #27210850