C₂×Q₃₂×C₅₃ #16810262

order	3392
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×Q₃₂×C₅₃

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	D₈
commutator, Frattini, cosocle	C₈	C₂³×C₅₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 87 points:

(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20);(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)

Conjugacy classes & character table

1166 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#28298981, #28298985, #28298999, …
maximal subgroup of	C₅₃×#1416, C₅₃×#1420, C₄×Q₃₂×C₅₃, …
maximal normal subgroup of	C₅₃×#1416, C₅₃×#1420, C₄×Q₃₂×C₅₃, …
maximal quotient of	C₂×C₅₃×#358, C₂×C₅₃×#366, C₄×Q₃₂×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3392, 188);

C2×Q32×C53 #16810262