D₄²×C₇₁ #17985676

order	4544
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₄²×C₇₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ G

428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 1136

A minimal faithful permutation representation — 4 generators on 79 points:

(1,2,3,4)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(2,4);(5,6,7,8);(6,8)

Conjugacy classes & character table

1775 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇₁×#1619, C₇₁×#1629, C₇₁×#1639, …
maximal normal subgroup of	C₇₁×#1619, C₇₁×#1629, C₇₁×#1639, …
maximal quotient of	D₄×C₂²⋊C₄×C₇₁, D₄×C₄⋊C₄×C₇₁, C₇₁×#2358, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 226);

D42×C71 #17985676