D₄×#13133123 #27304639

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₄×#13133123

Splittings

Sylow

C₅₃ ⋊ D₄×C₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Fitting	D₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂×D₄×C₅₃	C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	D₄×C₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

214 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 89 points:

(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)(58,59,60,64,61,65,68,74,62,66,69,75,71,77,80,84,63,67,70,76,72,78,81,85,73,79,82,86,83,87,88,89);(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)(55,57);(54,55)(56,57)

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90115);