C₁₃×7₋¹⁺² #17984972

order	4459
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₇²×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×7₋¹⁺²
unambiguous semidirect product	C₄₉⋊(C₇×C₁₃)
unambiguous semidirect product	(C₁₃×C₄₉)⋊C₇

Subgroups

	subgroup	G/H
center, socle	C₇×C₁₃	C₇²
commutator, Frattini, cosocle	C₇	C₇²×C₁₃
prime radical	C₇²×C₁₃	C₇
LCS	G ▷ C₇ ▷ C₁
UCS	C₁ ◁ C₇×C₁₃ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 62
transitive degree	μ_t(G) = 637

A minimal faithful permutation representation — 2 generators on 62 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,21,28,39,48,16,19,23,30,41,50,57,20,26,33,43,52,49,58,27,37,32,45,40,51,59,38,36,35,29,42,53,60,47,25,22,31,44,54,61,56,18,24,34,46,55,62);(15,18,25,36,37,26,19)(17,22,32,23,24,35,33)(21,29,30,31,43,34,45)(28,40,46,52,44,41,42)(39,49,53,55,50,51,54)(48,57,58,59,60,61,62)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₃×#14558877, C₁₃×#14558878, C₁₃×#14558879, …
maximal normal subgroup of	C₁₃×#14558877, C₁₃×#14558878, C₁₃×#14558879, …
maximal quotient of	C₁₃×C₇²⋊C₄₉, C₁₃×C₄₉⋊C₄₉, C₇×C₁₃×7₋¹⁺²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4459, 4);

C13×7−1+2 #17984972