C₂₃×Dic₄₉ #17985187

order	4508
isoclinism	isoclinic class of D₄₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₃×Dic₄₉
unambiguous semidirect product	C₄₉⋊(C₄×C₂₃)

Splittings

derived, Sylow	C₄₉ ⋊ C₄×C₂₃
Hall	C₂₃×C₄₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₃	D₄₉
commutator, hypocenter	C₄₉	C₄×C₂₃
Frattini	C₂×C₇	D₇×C₂₃
Fitting	C₂×C₂₃×C₄₉	C₂
socle, prime radical	C₂×C₇×C₂₃	D₇
cosocle	C₂×C₄₉	C₂×C₂₃
LCS	G ▷ C₄₉
UCS	C₁ ◁ C₂×C₂₃

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 4508

A minimal faithful permutation representation — 2 generators on 76 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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76);(24,26,25,27)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₃×(C₄₉⋊C₃)⋊C₄, C₃×C₂₃×Dic₄₉, C₂₃×Dic₁₄₇, …
maximal normal subgroup of	C₂₃×(C₄₉⋊C₃)⋊C₄, C₃×C₂₃×Dic₄₉, C₅×C₂₃×Dic₄₉
maximal quotient of	C₃×C₂₃×Dic₄₉, C₂₃×Dic₁₄₇, C₅×C₂₃×Dic₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4508, 2);

C23×Dic49 #17985187