Q₈×#41×C₂₉ #16817192

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₂₉ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

Q₈×#41×C₂₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₉
socle	C₂²×C₂₉	C₂⁵
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ G

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 45 points:

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

Conjugacy classes & character table

1450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8439, C₂₉×#17509, C₂₉×#17514, …
maximal normal subgroup of	C₂₉×#8439, C₂₉×#17509, C₂₉×#17514, …
maximal quotient of	C₄⋊C₄×#41×C₂₉, Q₈×C₂₉×#117, C₂₉×#34145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2210);

Q8×#41×C29 #16817192