D₄×SD₁₆×D₄₇ #27091486

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₄×SD₁₆×D₄₇

Splittings

Fitting	D₄×SD₁₆×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂×D₄×SD₁₆

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	D₄×SD₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	D₄²×D₄₇	C₂
hypercenter	D₄×SD₁₆	D₄₇
hypocenter	C₄₇	C₂×D₄×SD₁₆
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ D₄×SD₁₆

3226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 5 generators on 59 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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(53,56)(54,59)(57,58);(52,53,55,58)(54,56,59,57);(49,51);(48,49)(50,51)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1044776);

D4×SD16×D47 #27091486