C₄₉⋊C₈ #87621

order	392
isoclinism	isoclinic class of D₄₉
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₄₉⋊C₈

Splittings

derived, Sylow

C₄₉ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₄₉
commutator, hypocenter	C₄₉	C₈
Frattini	C₄×C₇	D₇
Fitting, cosocle	C₄×C₄₉	C₂
socle, prime radical	C₂×C₇	Dic₇
LCS	G ▷ C₄₉
UCS	C₁ ◁ C₄

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 392

A minimal faithful permutation representation — 2 generators on 57 points:

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

Polycyclic presentation:

2^3,7^2|a^2=b,b^2=c,d^7=e^6,d^a=d^6e,e^a=e^6

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₄₉, #11715985, C₂×C₄₉⋊C₈, …
maximal normal subgroup of	C₈×D₄₉, #11715985, C₂×C₄₉⋊C₈, …
maximal quotient of	C₄₉⋊C₁₆, C₂×C₄₉⋊C₈, C₃×C₄₉⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(392, 1);

C49⋊C8 #87621