D₄×Q₈×C₅₉ #16837083

order	3776
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₄×Q₈×C₅₉

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅₉
prime radical	C₂×D₄×C₅₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ G

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1888

A minimal faithful permutation representation — 4 generators on 71 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);(2,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12)

Conjugacy classes & character table

1475 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₉×#1627, C₅₉×#1637, C₅₉×#1649, …
maximal normal subgroup of	C₅₉×#1627, C₅₉×#1637, C₅₉×#1649, …
maximal quotient of	Q₈×C₂²⋊C₄×C₅₉, D₄×C₄⋊C₄×C₅₉, Q₈×C₄⋊C₄×C₅₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3776, 230);

D4×Q8×C59 #16837083