C₂⁴×Q₈×C₃₇ #17992407

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₃₇ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂⁴×Q₈×C₃₇

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃₇	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₃₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃₇ ◁ G

6264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 6 generators on 53 points:

(1,2)(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);(3,4);(5,6);(7,8);(9,10,11,12)(13,16,15,14);(9,13,11,15)(10,14,12,16)

Conjugacy classes & character table

2960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×#1446, C₃₇×#13922, C₃₇×#14336, …
maximal normal subgroup of	C₂×C₃₇×#1446, C₃₇×#13922, C₃₇×#14336, …
maximal quotient of	C₂⁴×C₄⋊C₄×C₃₇, C₂³×C₄×Q₈×C₃₇, C₂³×C₂²⋊Q₈×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2321);

C24×Q8×C37 #17992407