C₈×#41×C₂₉ #16816678

order	3712
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₈×#41×C₂₉

Subgroups

	subgroup	G/H
center	C₄×C₈×C₂₉	C₂²
commutator	C₂	C₂³×C₈×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂×#41×C₂₉	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₂₉ ◁ G

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4,5,6,7,8)(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);(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(9,11,12,15)(10,14,13,16)

Conjugacy classes & character table

2320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₈×C₂₉×#104, C₂₉×#8191, C₂₉×#8192, …
maximal normal subgroup of	C₈×C₂₉×#104, C₂₉×#8191, C₂₉×#8192, …
maximal quotient of	C₈×C₂₉×#117, C₄×C₈×D₄×C₂₉, C₄×C₈×Q₈×C₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1696);

C8×#41×C29 #16816678