C₂×D₅₁₃ #72210098

order	2052
automorphism	Aut(G) order 332424
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₂₆
direct product	C₂×D₅₁₃

Splittings

Fitting	C₂×C₁₉×C₂₇ ⋊ C₂
derived, Hall	C₁₉×C₂₇ ⋊ C₂²
Sylow	C₂₇ ⋊ C₂×D₁₉
Sylow	C₁₉ ⋊ C₂×D₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₁₃
commutator, cosocle, hypocenter	C₁₉×C₂₇	C₂²
Frattini	C₉	C₂×D₅₇
Fitting	C₂×C₁₉×C₂₇	C₂
socle	C₂×C₃×C₁₉	D₉
LCS	G ▷ C₁₉×C₂₇
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 1026

A minimal faithful permutation representation — 2 generators on 48 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);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₁₅₃₉, D₂₀₅₂, C₂×D₂₅₆₅, …
maximal normal subgroup of	D₂₀₅₂
maximal quotient of	C₂×D₁₅₃₉, D₂₀₅₂, C₂×D₂₅₆₅, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((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), (4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40));

C2×D513 #72210098