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#213

order	48
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂×D₃×D₄ (order 96) · Inn(G) = #67 · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #72215117
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₃×D₄ ⋊ C₂
Sylow	C₃ ⋊ D₈

Subgroups

	subgroup	G/H
center	C₂	#67
commutator, cosocle	C₃×C₄	C₂²
Frattini	C₄	C₂×D₃
Fitting	C₃×D₄	C₂
socle	C₂×C₃	D₄
hypercenter	D₄	D₃
hypocenter	C₃	D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Polycyclic presentation:

2^4,3|c^2=d,b^a=bc,c^a=cd,e^a=e^2,c^b=cd

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	1	2	2	2	4	4	4	4	6	6	12
order	1	2	3	4	6	2	6	6	12	8	8	2
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	1	1	1	1	1	-1	-1	-1	1	1	1	-1
χ₄	1	1	1	1	1	-1	-1	-1	1	-1	-1	1
χ₅	2	2	2	-2	2	0	0	0	-2	0	0	0
χ₆	2	2	-1	2	-1	2	-1	-1	-1	0	0	0
χ₇	2	2	-1	2	-1	-2	1	1	-1	0	0	0
χ₈	2	2	-1	-2	-1	0	$- i \sqrt{3}$	$i \sqrt{3}$	1	0	0	0
χ₉	2	2	-1	-2	-1	0	$i \sqrt{3}$	$- i \sqrt{3}$	1	0	0	0
χ₁₀	2	-2	2	0	-2	0	0	0	0	$\sqrt{2}$	$- \sqrt{2}$	0
χ₁₁	2	-2	2	0	-2	0	0	0	0	$- \sqrt{2}$	$\sqrt{2}$	0
χ₁₂	4	-4	-2	0	2	0	0	0	0	0	0	0

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

Relationships

maximal subgroup of	D₃×D₈, #911, #914, …
maximal normal subgroup of	D₃×D₈, #911, #914, …
maximal quotient of	#807, #809, #826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 15);