#66900

order	352
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₁₁ ⋊ #101

Subgroups

	subgroup	G/H
center	C₂	C₂²⋊Dic₁₁
commutator, prime radical	C₂²×C₁₁	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₁
Fitting	C₂×Q₈×C₁₁	C₂
socle	C₂×C₁₁	C₂²⋊C₄
cosocle	C₂×C₄×C₁₁	C₂²
hypercenter	C₂×Q₈	D₁₁
hypocenter	C₁₁	#101
LCS	G ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,4,6,8,10,11,9,7,5,3)(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,18,16,19,24,27)(14,20,21,26,22,17,23,25)

Polycyclic presentation:

2^5,11|a^2=c,b^2=e,c^2=e,b^a=bd,d^a=de,f^a=f^10,c^b=ce

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622115, #10622116, #10622180, …
maximal normal subgroup of	#10622115, #10622116, #10622180, …
maximal quotient of	#10622166, #10622168, #10622181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 42);