D₄×#13133143 #27399760

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₄×#13133143

Splittings

Fitting	D₄×M₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ D₄×#104

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	D₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	D₄×#11719945	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	D₄×M₁₆	D₅₃
hypocenter	C₅₃	D₄×#104
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ D₄×M₁₆

1608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 65 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)(58,60)(59,62)(61,63)(64,65);(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)(58,59,61,64,62,60,65,63);(55,57);(54,55)(56,57)

Conjugacy classes & character table

1370 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 185236);

D4×#13133143 #27399760