Q₈×#41×D₄₇ #27123346

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

Q₈×#41×D₄₇

Splittings

Fitting	Q₈×#41×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂×Q₈×#41

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	Q₈×#41×C₄₇	C₂
prime radical	C₂×#41×D₄₇	C₂²
hypercenter	Q₈×#41	D₄₇
hypocenter	C₄₇	C₂×Q₈×#41
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ Q₈×#41

7522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 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);(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)(56,58,59,62)(57,61,60,63);(57,60)(61,63);(56,57)(58,61)(59,60)(62,63);(48,50,51,55)(49,52,53,54);(48,49,51,53)(50,54,55,52)

Conjugacy classes & character table

1250 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076636);

Q8×#41×D47 #27123346