Dic₂₈₄₉ #26037518

order	11396
automorphism	Aut(G) order 12307680
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₈₄₉
unambiguous semidirect product	C₇⋊Dic₄₀₇
unambiguous semidirect product	C₁₁⋊Dic₂₅₉

Splittings

derived, Hall	C₇×C₁₁×C₃₇ ⋊ C₄
Hall	C₁₁×C₃₇ ⋊ Dic₇
Hall	C₇×C₃₇ ⋊ Dic₁₁
Hall	C₇×C₁₁ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₇₇
Sylow	C₁₁ ⋊ Dic₂₅₉
Sylow	C₇ ⋊ Dic₄₀₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₈₄₉
commutator, hypocenter	C₇×C₁₁×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₁×C₃₇	C₂
LCS	G ▷ C₇×C₁₁×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 11396

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

Polycyclic presentation:

2,7,11,37,2|a^2=e,b^a=b^6,c^a=c^10,d^a=d^36

Conjugacy classes & character table

2852 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₈₄₉, Dic₈₅₄₇, C₁₁⋊#15690918, …
maximal normal subgroup of	C₃×Dic₂₈₄₉, C₁₁⋊#15690918, C₁₁⋊(ΓL(1,𝔽₈)⋊Dic₃₇), …
maximal quotient of	C₃×Dic₂₈₄₉, Dic₈₅₄₇, Dic₁₄₂₄₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11396, 8);

Dic2849 #26037518