C₁₃×#12772 #15699111

order	3328
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#12772

Subgroups

	subgroup	G/H
center	C₄×C₁₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×#41	C₂³×C₁₃
socle	C₂×C₁₃	#1790
prime radical	C₁₃×#3529	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#41 ◁ G

760 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,26)(17,30,18,23)(19,24)(20,25)(21,27,37,32)(22,31,38,33)(28,35)(29,39)(34,41)(36,42,40,43)(44,45);(14,16)(15,21)(17,27)(18,32)(19,28)(20,29)(22,35)(23,39)(24,40)(25,37)(26,30)(31,42)(33,43)(34,44)(36,41)(38,45);(14,17)(15,22)(16,27)(18,20)(19,33)(21,35)(23,24)(25,38)(26,36)(28,43)(29,32)(30,41)(31,34)(37,45)(39,40)(42,44)

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5760);