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C₂²×Dic₂₂ #67031

order	352
isoclinism	isoclinic class of Dic₂₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×Dic₂₂

Splittings

C₁₁ ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₁
commutator, cosocle	C₂×C₁₁	C₂⁴
Frattini	C₂	C₂³×D₁₁
Fitting	C₂²×C₄×C₁₁	C₂
socle, prime radical	C₂³×C₁₁	C₂²
hypercenter	C₂²×C₄	D₁₁
hypocenter	C₁₁	C₂²×Q₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 352

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622259, #10622269, #10622286, …
maximal normal subgroup of	#10622259, #10622269, #10622286, …
maximal quotient of	C₂×C₄×Dic₂₂, C₂×#66922, C₂×#66923, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 173);