C₂×C₅²×C₆₇ #16809774

order	3350
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 31680
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₅

Aliases

abelian

C₂×C₅²×C₆₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 3350

A minimal faithful permutation representation — 2 generators on 79 points:

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

Conjugacy classes & character table

3350 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₂×C₅²⋊C₃×C₆₇⋊C₃, C₂×C₅²×C₆₇⋊C₁₁
socle of	C₂×C₅²⋊C₃×C₆₇⋊C₃, C₂×C₅²×C₆₇⋊C₁₁
maximal subgroup of	C₂×C₅²×C₇×C₆₇, C₅²×D₇×C₆₇, C₂×C₅²×C₁₁×C₆₇, …
maximal normal subgroup of	C₂×C₅²×C₇×C₆₇, C₂×C₅²×C₁₁×C₆₇, C₂×C₅²×C₆₇⋊C₁₁, …
maximal quotient of	C₂×C₅²×C₇×C₆₇, C₅²×D₇×C₆₇, C₂×C₅²×C₁₁×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3350, 5);

C2×C52×C67 #16809774