#66883

order	352
isoclinism	isoclinic class of #4314
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₁₁ ⋊ #108

Subgroups

	subgroup	G/H
center	C₄	D₄₄
commutator	C₄×C₁₁	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₁
Fitting	C₂×C₈×C₁₁	C₂
socle	C₂×C₁₁	C₄⋊C₄
prime radical	C₂²×C₁₁	Q₈
cosocle	C₂×C₄×C₁₁	C₂²
hypercenter	C₂×C₈	D₁₁
hypocenter	C₁₁	#108
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

30 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)(12,14,16,23,17,24,15,22)(13,18,19,21,20,26,25,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,19,17,20,16,25)(14,21,22,26,24,27,23,18)

Polycyclic presentation:

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

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622125, #10622131, #10622144, …
maximal normal subgroup of	#10622125, #10622131, #10622144, …
maximal quotient of	#10622095, #10622096, #10622190, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 25);