C₂²×#10621729 #72229209

order	2704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₁₃

Aliases

direct product

C₂²×#10621729

Splittings

Fitting, socle	C₂²×C₁₃² ⋊ C₄
derived, Sylow	C₁₃² ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#10621729
commutator, hypocenter	C₁₃²	C₂²×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₁₃²	C₄
prime radical	C₂²×C₁₃⋊D₁₃	C₂
cosocle	C₁₃⋊D₁₃	C₂³
LCS	G ▷ C₁₃²
UCS	C₁ ◁ C₂²

204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 3 generators on 30 points:

(1,11,2,3,5,7,8,13,12,9,6,4,10)(14,15,19,20,25,17,23,26,22,21,24,16,18)(29,30);(2,3,4,6)(5,8,9,13)(7,11,12,10)(15,17,18,22)(16,20,19,24)(21,23,25,26);(27,28)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#14523146, C₂²×D₁₃⋊#261, C₂×#72214059, …
maximal normal subgroup of	C₂²×#14523146, C₂²×D₁₃⋊#261, C₂×#72214059, …
maximal quotient of	C₂×#72214162, #72259202, #72259233, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,11,2,3,5,7,8,13,12,9,6,4,10)(14,15,19,20,25,17,23,26,22,21,24,16,18)(29,30), (2,3,4,6)(5,8,9,13)(7,11,12,10)(15,17,18,22)(16,20,19,24)(21,23,25,26), (27,28));

C22×#10621729 #72229209