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Q₁₆×Dic₇ #89955

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Q₁₆×Dic₇

Splittings

C₇ ⋊ C₄×Q₁₆

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	C₄×Q₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726960, #11726961, #11726962, …
maximal normal subgroup of	#11726960, #11726961, #11726962, …
maximal quotient of	Dic₇×#103, #11727978, #11727990, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 717);