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C₂²×C₃×C₄ #242

order	48
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#5934 (order 384) · Inn(G) = C₁ · Out(G) = C₂×#5934
holomorph	G ⋊ Aut(G) = D₃×#72209832
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂³

Aliases

abelian	C₂²×C₃×C₄
multiplicative	(ℤ/104)ˣ, (ℤ/105)ˣ, (ℤ/112)ˣ, (ℤ/140)ˣ, (ℤ/144)ˣ, (ℤ/156)ˣ, (ℤ/180)ˣ, (ℤ/210)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 11 points:

(1,2,3,4)(9,10,11);(5,6);(7,8)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

center of	C₃×#336, C₃×#376, C₃×C₄×C₂²⋊C₄, …
derived subgroup of	#67448, #67449, #67450, …
Frattini subgroup of	#338⋊C₉, #10588128, #10588129, …
Fitting subgroup of	#830, C₃⋊#95, C₂²×C₃⋊C₈, …
maximal subgroup of	#830, C₃⋊#95, C₃×#95, …
maximal normal subgroup of	#830, C₃⋊#95, C₃×#95, …
maximal quotient of	C₂×C₃×C₄², C₂×C₃×C₂²⋊C₄, C₂×C₃×C₄⋊C₄, …
automorphism group of	C₈×C₁₃, C₃×C₅×C₇, C₇×C₁₆, …
outer automorphism group of	C₈×C₁₃, C₃×C₅×C₇, C₇×C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 44);