Dic₂₇₃₇ #24926504

order	10948
automorphism	Aut(G) order 11561088
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₃₇
unambiguous semidirect product	C₇⋊Dic₃₉₁
unambiguous semidirect product	C₂₃⋊Dic₁₁₉

Splittings

derived, Hall	C₇×C₁₇×C₂₃ ⋊ C₄
Hall	C₁₇×C₂₃ ⋊ Dic₇
Hall	C₇×C₂₃ ⋊ Dic₁₇
Hall	C₇×C₁₇ ⋊ Dic₂₃
Sylow	C₂₃ ⋊ Dic₁₁₉
Sylow	C₁₇ ⋊ Dic₁₆₁
Sylow	C₇ ⋊ Dic₃₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₇₃₇
commutator, hypocenter	C₇×C₁₇×C₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₂₃	C₂
LCS	G ▷ C₇×C₁₇×C₂₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 10948

A minimal faithful permutation representation — 2 generators on 51 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);(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)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2,7,17,23,2|a^2=e,b^a=b^6,c^a=c^16,d^a=d^22

Conjugacy classes & character table

2740 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₇₃₇, Dic₈₂₁₁, ΓL(1,𝔽₈)⋊Dic₃₉₁, …
maximal normal subgroup of	C₃×Dic₂₇₃₇, ΓL(1,𝔽₈)⋊Dic₃₉₁, Dic₅₄₇₄
maximal quotient of	C₃×Dic₂₇₃₇, Dic₈₂₁₁, Dic₁₃₆₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10948, 8);

Dic2737 #24926504