C₂×C₇×C₃⋊D₃×C₅₃ #27211329

order	13356
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₅₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₇×C₃⋊D₃×C₅₃
unambiguous semidirect product	(C₂×C₃)⋊(D₃×C₇×C₅₃)

Splittings

Fitting, socle	C₂×C₃²×C₇×C₅₃ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂²×C₇×C₅₃
Hall	C₃²×C₇×C₅₃ ⋊ C₂²
Hall	C₃²×C₅₃ ⋊ C₂²×C₇
Hall	C₃²×C₇ ⋊ C₂²×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇×C₅₃	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂²×C₇×C₅₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇×C₅₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₇×C₅₃

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 6678

A minimal faithful permutation representation — 3 generators on 68 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)(62,63)(65,66);(64,65,66)(67,68);(61,62,63)

Conjugacy classes & character table

4452 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 122);

C2×C7×C3⋊D3×C53 #27211329