D₅₄₅₃ #24926314

order	10906
automorphism	Aut(G) order 23556960
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₄₅₃
unambiguous semidirect product	C₇⋊D₇₇₉
unambiguous semidirect product	C₁₉⋊D₂₈₇
unambiguous semidirect product	C₄₁⋊D₁₃₃

Splittings

Fitting, socle, derived, Hall	C₇×C₁₉×C₄₁ ⋊ C₂
Hall	C₁₉×C₄₁ ⋊ D₇
Hall	C₇×C₄₁ ⋊ D₁₉
Hall	C₇×C₁₉ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₁₃₃
Sylow	C₁₉ ⋊ D₂₈₇
Sylow	C₇ ⋊ D₇₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₉×C₄₁	C₂
LCS	G ▷ C₇×C₁₉×C₄₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 5453

A minimal faithful permutation representation — 2 generators on 67 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(2,7)(3,6)(4,5)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,67)(29,66)(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:

2,7,19,41|b^a=b^6,c^a=c^18,d^a=d^40

Conjugacy classes & character table

2728 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₁×D₁₃₃, D₁₉×D₂₈₇, D₇×D₇₇₉, …
maximal normal subgroup of	D₄₁×D₁₃₃, D₁₉×D₂₈₇, D₇×D₇₇₉, …
maximal quotient of	Dic₅₄₅₃, C₂×D₅₄₅₃, C₃×D₅₄₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10906, 8);

D5453 #24926314