Q₃₂×D₂₁₂ #27540821

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

Q₃₂×D₂₁₂

Splittings

Fitting	C₄×Q₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ D₄×Q₃₂

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₂₁₂	D₈
hypercenter	C₄×Q₃₂	D₅₃
hypocenter	C₅₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

664 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 159 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 — 4 generators on 89 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)(54,55,56,57);(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)(55,57);(58,60,63,71)(59,64,67,81)(61,73,76,89)(62,78,74,70)(65,83,84,88)(66,85,77,80)(68,86,87,72)(69,82,79,75);(58,59,63,67)(60,68,71,87)(61,72,76,86)(62,77,74,66)(64,79,81,69)(65,82,84,75)(70,88,78,83)(73,85,89,80)

Conjugacy classes & character table

1199 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326297);