D₄⋊#13133123 #27271530

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

D₄⋊#13133123

Splittings

Sylow

C₅₃ ⋊ D₄⋊C₃₂

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	D₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²⋊C₁₆
prime radical	C₂×D₄×C₅₃	C₁₆
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	D₄⋊C₃₂
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ D₄×C₁₆

138 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 2 generators on 93 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(55,58)(56,61)(59,60);(2,3)(4,5)(6,7)(8,9)(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)(56,58)(57,60)(59,61)(62,63,64,68,65,69,72,78,66,70,73,79,75,81,84,88,67,71,74,80,76,82,85,89,77,83,86,90,87,91,92,93)

Polycyclic presentation:

2^8,53|a^2=d,c^2=e,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ce,i^a=i^52,c^b=ce

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57006);