D₄×Dic₄₂₄ #27540822

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

Aliases

direct product

D₄×Dic₄₂₄

Splittings

Sylow

C₅₃ ⋊ D₄×Q₃₂

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	D₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄×C₅₃	D₈
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

664 conjugacy classes of subgroups.

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)(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)(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);(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)(58,59,63,67)(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);(55,57);(54,55)(56,57)

Conjugacy classes & character table

2135 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326298);

D4×Dic424 #27540822