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C₂²×Dic₁₄ #6542

order	224
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) order 129024
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×Dic₁₄

Splittings

C₇ ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₇
commutator, cosocle	C₂×C₇	C₂⁴
Frattini	C₂	C₂³×D₇
Fitting	C₂²×C₄×C₇	C₂
socle, prime radical	C₂³×C₇	C₂²
hypercenter	C₂²×C₄	D₇
hypocenter	C₇	C₂²×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#89418, #89428, #89445, …
maximal normal subgroup of	#89418, #89428, #89445, …
maximal quotient of	C₂×C₄×Dic₁₄, C₂×#6432, C₂×#6433, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 174);