C₄×C₈₉ #67062

order	356
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₈×C₁₁ (order 176) · Inn(G) = C₁ · Out(G) = C₂×C₈×C₁₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₅₆
abelian	C₄×C₈₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₈₉
socle, prime radical	C₂×C₈₉	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 356

A minimal faithful permutation representation — 1 generator on 93 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)

Conjugacy classes & character table

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

Relationships

center of	M₁₆×C₈₉, #41×C₈₉, #104×C₈₉, …
derived subgroup of	#13083559, D₇₁₂, #13083561, …
Fitting subgroup of	#10623494, Dic₁₇₈, C₄×D₈₉, …
maximal subgroup of	#10623494, C₈×C₈₉, Dic₁₇₈, …
maximal normal subgroup of	#10623494, C₈×C₈₉, Dic₁₇₈, …
maximal quotient of	C₈×C₈₉, C₂×C₄×C₈₉, Dic₃×C₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(356, 2);

C4×C89 #67062