C₁₃×#2108 #13111166

order	1664
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#2108

Subgroups

	subgroup	G/H
center	C₂×C₁₃×C₁₆	C₂²
commutator	C₂	C₂×C₄×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂²×C₈
prime radical	C₂³×C₁₃	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃×C₁₆ ◁ G

184 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 49
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,21,25,22,26,31,37,23,27,32,38,33,39,43,47)(20,24,29,35,30,36,42,46,28,34,40,44,41,45,48,49);(14,15,16,17)(18,20)(19,24)(21,29)(22,30)(23,28)(25,35)(26,36)(27,34)(31,42)(32,40)(33,41)(37,46)(38,44)(39,45)(43,48)(47,49);(20,28)(24,34)(29,40)(30,41)(35,44)(36,45)(42,48)(46,49)

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#15750614, #15750617, #15774577, …
maximal subgroup of	C₁₃×#7469, C₁₃×#7470, C₁₃×#12026, …
maximal normal subgroup of	C₁₃×#7469, C₁₃×#7470, C₁₃×#12026, …
maximal quotient of	C₁₃×#11399, C₁₃×#11401, C₁₃×C₁₆×M₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 840);