C₂³×C₁₆×C₂₉ #16817118

order	3712
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 2408448
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁶

Aliases

abelian	C₂³×C₁₆×C₂₉
multiplicative	(ℤ/8024)ˣ, (ℤ/11328)ˣ, (ℤ/12036)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂⁴×C₂₉
socle, prime radical	C₂⁴×C₂₉	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 4 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(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);(17,18);(19,20);(21,22)

Conjugacy classes & character table

3712 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂₉×#1374, C₂₉×#12545, C₂×C₂₉×#1379, …
maximal normal subgroup of	C₂×C₂₉×#1374, C₂₉×#12545, C₂×C₂₉×#1379, …
maximal quotient of	C₂²×C₂₉×C₂²⋊C₁₆, C₂²×C₂₉×C₄⋊C₁₆, C₂×C₂₉×#2162, …
automorphism group of	C₈×C₁₇×C₅₉, C₃×C₅₉×C₆₄
outer automorphism group of	C₈×C₁₇×C₅₉, C₃×C₅₉×C₆₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2136);

C23×C16×C29 #16817118