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C₂²⋊C₄×D₁₃ #88134

order	416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂²⋊C₄×D₁₃

Splittings

Fitting	C₁₃×C₂²⋊C₄ ⋊ C₂
Sylow	C₁₃ ⋊ C₂×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₃
commutator	C₂×C₁₃	C₂²×C₄
Fitting	C₁₃×C₂²⋊C₄	C₂
socle, cosocle	C₂²×C₁₃	C₂³
prime radical	C₂³×D₁₃	C₂
hypercenter	C₂²⋊C₄	D₁₃
hypocenter	C₁₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718238, D₁₃×#99, C₂²⋊C₄×#261, …
maximal normal subgroup of	#11718238, D₁₃×#99, C₂²⋊C₄×#261, …
maximal quotient of	#11718307, D₁₃×#95, #11718325, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 101);