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C₄×D₃ #64

order	24
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×D₃ (order 24) · Inn(G) = D₃ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #10596675
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₃

Splittings

Fitting	C₃×C₄ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₃
commutator, hypocenter	C₃	C₂×C₄
Frattini	C₂	C₂×D₃
Fitting	C₃×C₄	C₂
socle, cosocle	C₂×C₃	C₂²
prime radical	C₂×D₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 7 points:

(2,3);(1,2,3)(4,5,6,7)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	2	3	1	2	3	1	2	3	1	2	3
order	1	3	2	4	12	4	2	6	2	4	12	4
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	-1	1	1	-1	1	1	-1	1	1	-1
χ₃	2	-1	0	2	-1	0	2	-1	0	2	-1	0
χ₄	1	1	1	$i$	$i$	$i$	-1	-1	-1	$- i$	$- i$	$- i$
χ₅	1	1	-1	$i$	$i$	$- i$	-1	-1	1	$- i$	$- i$	$i$
χ₆	2	-1	0	$2 i$	$- i$	0	-2	1	0	$- 2 i$	$i$	0
χ₇	1	1	1	-1	-1	-1	1	1	1	-1	-1	-1
χ₈	1	1	-1	-1	-1	1	1	1	-1	-1	-1	1
χ₉	2	-1	0	-2	1	0	2	-1	0	-2	1	0
χ₁₀	1	1	1	$- i$	$- i$	$- i$	-1	-1	-1	$i$	$i$	$i$
χ₁₁	1	1	-1	$- i$	$- i$	$i$	-1	-1	1	$i$	$i$	$- i$
χ₁₂	2	-1	0	$- 2 i$	$i$	0	-2	1	0	$2 i$	$- i$	0

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup of	D₃×C₈, ΓL(1,𝔽₂₅), C₂×C₄×D₃, …
maximal normal subgroup of	D₃×C₈, ΓL(1,𝔽₂₅), C₂×C₄×D₃, …
maximal quotient of	D₃×C₈, ΓL(1,𝔽₂₅), C₄×Dic₃, …
automorphism group of	C₂²×C₅, C₅×D₃
outer automorphism group of	C₂²×C₅, C₅×Q₈, C₃×C₁₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24, 5);