Q₈×#13094048 #27096848

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

Q₈×#13094048

Splittings

Fitting	Q₈×D₈×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ Q₈×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	Q₈×D₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×#13094048	C₂²
hypercenter	Q₈×D₈	D₄₇
hypocenter	C₄₇	Q₈×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈×D₈

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 63 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)(57,61)(60,62);(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)(57,60)(58,63)(61,62);(56,57)(58,60)(59,61)(62,63);(48,50,51,55)(49,52,53,54);(48,49,51,53)(50,54,55,52)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1050138);

Q8×#13094048 #27096848