C₂×C₄×C₁₃×C₃₂ #15699885

order	3328
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₄

Aliases

abelian	C₂×C₄×C₁₃×C₃₂
multiplicative	(ℤ/6784)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
socle, prime radical	C₂³×C₁₃	C₂×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 3328

A minimal faithful permutation representation — 3 generators on 51 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)(39,40,41,42,43,44,45,46,47,48,49,50,51);(33,34,35,36);(37,38)

Conjugacy classes & character table

3328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄×C₁₃×C₆₄, C₂×C₈×C₁₃×C₃₂, C₄²×C₁₃×C₃₂, …
maximal normal subgroup of	C₂×C₄×C₁₃×C₆₄, C₂×C₈×C₁₃×C₃₂, C₄²×C₁₃×C₃₂, …
maximal quotient of	C₂×C₄×C₁₃×C₆₄, C₂×C₈×C₁₃×C₃₂, C₄²×C₁₃×C₃₂, …
automorphism group of	C₅₃×C₁₂₈
outer automorphism group of	C₅₃×C₁₂₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6534);

C2×C4×C13×C32 #15699885