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C₂×Dic₂₈ #6468

order	224
isoclinism	isoclinic class of #1132
automorphism	Aut(G) order 10752
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

C₂×Dic₂₈

Splittings

C₇ ⋊ C₂×Q₁₆

Subgroups

	subgroup	G/H
center	C₂²	D₂₈
commutator, cosocle	C₄×C₇	C₂³
Frattini	C₄	C₂²×D₇
Fitting	C₂×C₇×C₈	C₂
socle, prime radical	C₂²×C₇	D₄
hypercenter	C₂×C₈	D₇
hypocenter	C₇	C₂×Q₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#89287, #89291, #89298, …
maximal normal subgroup of	#89287, #89291, #89298, …
maximal quotient of	#89453, #89454, C₄×Dic₂₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 100);