D₁₂×C₄₉ #11924116

order	1176
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₁₂×C₄₉
unambiguous semidirect product	C₄⋊(D₃×C₄₉)

Splittings

Fitting	C₃×C₄×C₄₉ ⋊ C₂
Hall	C₃×C₄₉ ⋊ D₄
Sylow	C₃ ⋊ D₄×C₄₉

Subgroups

	subgroup	G/H
center	C₂×C₄₉	C₂×D₃
commutator	C₂×C₃	C₂²×C₄₉
Frattini	C₂×C₇	C₂×D₃×C₇
Fitting	C₃×C₄×C₄₉	C₂
socle, cosocle	C₂×C₃×C₇	C₂²×C₇
prime radical	C₇×D₁₂	C₇
hypercenter	C₄×C₄₉	D₃
hypocenter	C₃	D₄×C₄₉
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄₉ ◁ C₄×C₄₉

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 588

A minimal faithful permutation representation — 2 generators on 56 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);(2,4)(6,7)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 31);

D12×C49 #11924116