D₄×Q₈×C₆₁ #16839443

order	3904
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₄×Q₈×C₆₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₆₁
prime radical	C₂×D₄×C₆₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ G

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 4 generators on 73 points:

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

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#30598437, #30598439, #30598455, …
maximal subgroup of	C₆₁×#1627, C₆₁×#1637, C₆₁×#1649, …
maximal normal subgroup of	C₆₁×#1627, C₆₁×#1637, C₆₁×#1649, …
maximal quotient of	Q₈×C₂²⋊C₄×C₆₁, D₄×C₄⋊C₄×C₆₁, Q₈×C₄⋊C₄×C₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3904, 230);

D4×Q8×C61 #16839443