C₄×C₂₉×#108 #16815491

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₄×C₂₉×#108

Subgroups

	subgroup	G/H
center	C₄²×C₂₉	D₄
commutator	C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×C₄⋊C₄
prime radical	C₂³×C₂₉	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 49 points:

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

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#7172, C₂₉×#7173, C₂₉×#7176, …
maximal normal subgroup of	C₂₉×#7172, C₂₉×#7173, C₂₉×#7176, …
maximal quotient of	C₄×C₂₉×#334, C₄×C₂₉×#335, C₂₉×#8273, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 509);

C4×C29×#108 #16815491