D24:C2 - GroupNames

G = D₂₄:C₂ order 96 = 2⁵·3

5^th semidirect product of D₂₄ and C₂ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₄:₅C₂, Q₁₆:₃S₃, Dic₃ o Q₁₆, D₆.₃D₄, C₈.₁₀D₆, Q₈.₁₀D₆, C₂₄.₈C₂², C₁₂.₁₀C₂³, Dic₃.₁₄D₄, D₁₂.₅C₂², (S₃xC₈):₃C₂, C₃:₄(C₄oD₈), (C₃xQ₁₆):₃C₂, C₂.₂₄(S₃xD₄), C₆.₃₆(C₂xD₄), C₃:C₈.₈C₂², Q₈:₂S₃:₄C₂, Q₈:₃S₃:₃C₂, C₄.₁₀(C₂²xS₃), (C₃xQ₈).₅C₂², (C₄xS₃).₁₂C₂², SmallGroup(96,126)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₂₄:C₂

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₄xS₃ — Q₈:₃S₃ — D₂₄:C₂

Lower central

C₃ — C₆ — C₁₂ — D₂₄:C₂

Upper central

C₁ — C₂ — C₄ — Q₁₆

Generators and relations for D₂₄:C₂
G = < a,b,c | a²⁴=b²=c²=1, bab=a^-1, cac=a¹⁷, cbc=a⁴b >

Subgroups: 162 in 62 conjugacy classes, 27 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₈, C₈, C₂xC₄, D₄, Q₈, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₃:C₈, C₂₄, C₄xS₃, C₄xS₃, D₁₂, D₁₂, C₃xQ₈, C₄oD₈, S₃xC₈, D₂₄, Q₈:₂S₃, C₃xQ₁₆, Q₈:₃S₃, D₂₄:C₂
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂²xS₃, C₄oD₈, S₃xD₄, D₂₄:C₂

Character table of D₂₄:C₂

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	6	8A	8B	8C	8D	12A	12B	12C	24A	24B
size	1	1	6	12	12	2	2	3	3	4	4	2	2	2	6	6	4	8	8	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	-1	-1	-1	1	1	1	-1	-1	1	-1	-1	1	1	linear of order 2
ρ₃	1	1	1	-1	1	1	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	1	-1	-1	linear of order 2
ρ₄	1	1	1	1	-1	1	1	1	1	-1	1	1	-1	-1	-1	-1	1	1	-1	-1	-1	linear of order 2
ρ₅	1	1	-1	-1	-1	1	1	-1	-1	1	1	1	1	1	-1	-1	1	1	1	1	1	linear of order 2
ρ₆	1	1	-1	-1	1	1	1	-1	-1	-1	1	1	-1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₇	1	1	-1	1	-1	1	1	-1	-1	1	-1	1	-1	-1	1	1	1	-1	1	-1	-1	linear of order 2
ρ₈	1	1	1	-1	-1	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	1	1	linear of order 2
ρ₉	2	2	0	0	0	-1	2	0	0	-2	2	-1	-2	-2	0	0	-1	-1	1	1	1	orthogonal lifted from D₆
ρ₁₀	2	2	0	0	0	-1	2	0	0	-2	-2	-1	2	2	0	0	-1	1	1	-1	-1	orthogonal lifted from D₆
ρ₁₁	2	2	2	0	0	2	-2	-2	-2	0	0	2	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	-2	0	0	2	-2	2	2	0	0	2	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	0	0	0	-1	2	0	0	2	2	-1	2	2	0	0	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₄	2	2	0	0	0	-1	2	0	0	2	-2	-1	-2	-2	0	0	-1	1	-1	1	1	orthogonal lifted from D₆
ρ₁₅	2	-2	0	0	0	2	0	2i	-2i	0	0	-2	√2	-√2	-√-2	√-2	0	0	0	-√2	√2	complex lifted from C₄oD₈
ρ₁₆	2	-2	0	0	0	2	0	-2i	2i	0	0	-2	-√2	√2	-√-2	√-2	0	0	0	√2	-√2	complex lifted from C₄oD₈
ρ₁₇	2	-2	0	0	0	2	0	-2i	2i	0	0	-2	√2	-√2	√-2	-√-2	0	0	0	-√2	√2	complex lifted from C₄oD₈
ρ₁₈	2	-2	0	0	0	2	0	2i	-2i	0	0	-2	-√2	√2	√-2	-√-2	0	0	0	√2	-√2	complex lifted from C₄oD₈
ρ₁₉	4	4	0	0	0	-2	-4	0	0	0	0	-2	0	0	0	0	2	0	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₀	4	-4	0	0	0	-2	0	0	0	0	0	2	2√2	-2√2	0	0	0	0	0	√2	-√2	orthogonal faithful
ρ₂₁	4	-4	0	0	0	-2	0	0	0	0	0	2	-2√2	2√2	0	0	0	0	0	-√2	√2	orthogonal faithful

Smallest permutation representation of D₂₄:C₂
►On 48 points

Generators in S₄₈

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)

(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(45 48)(46 47)

(1 29)(2 46)(3 39)(4 32)(5 25)(6 42)(7 35)(8 28)(9 45)(10 38)(11 31)(12 48)(13 41)(14 34)(15 27)(16 44)(17 37)(18 30)(19 47)(20 40)(21 33)(22 26)(23 43)(24 36)

G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,48)(46,47), (1,29)(2,46)(3,39)(4,32)(5,25)(6,42)(7,35)(8,28)(9,45)(10,38)(11,31)(12,48)(13,41)(14,34)(15,27)(16,44)(17,37)(18,30)(19,47)(20,40)(21,33)(22,26)(23,43)(24,36)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,48)(46,47), (1,29)(2,46)(3,39)(4,32)(5,25)(6,42)(7,35)(8,28)(9,45)(10,38)(11,31)(12,48)(13,41)(14,34)(15,27)(16,44)(17,37)(18,30)(19,47)(20,40)(21,33)(22,26)(23,43)(24,36) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(45,48),(46,47)], [(1,29),(2,46),(3,39),(4,32),(5,25),(6,42),(7,35),(8,28),(9,45),(10,38),(11,31),(12,48),(13,41),(14,34),(15,27),(16,44),(17,37),(18,30),(19,47),(20,40),(21,33),(22,26),(23,43),(24,36)]])

D₂₄:C₂ is a maximal subgroup of
D₄₈:C₂ D₆.₂D₈ Q₃₂:S₃ D₄₈:₅C₂ D₁₂.₃₀D₄ S₃xC₄oD₈ D₈:₁₅D₆ D₂₄:C₂² C₂₄.C₂³ D₇₂:₅C₂ D₆.₃D₁₂ D₂₄:₅S₃ D₁₂.₁₃D₆ D₁₂.₁₄D₆ C₂₄.₂₈D₆ Dic₃.₅S₄ D₁₂₀:₅C₂ D₂₄:₅D₅ Dic₁₀.₂₇D₆ D₁₂.D₁₀ D₁₂₀:₈C₂
D₂₄:C₂ is a maximal quotient of
Dic₃:₇SD₁₆ Q₈.₃Dic₆ Q₈:C₄:S₃ C₄:C₄.₁₅₀D₆ D₆:₂SD₁₆ Q₈:₄D₁₂ D₆:C₈.C₂ D₁₂.₁₂D₄ Dic₃:₅D₈ C₈.₆Dic₆ C₈.₂₇(C₄xS₃) D₆:₂D₈ C₂.D₈:S₃ D₁₂.₂Q₈ Dic₃xQ₁₆ (C₂xQ₁₆):S₃ D₁₂.₁₇D₄ D₆:₃Q₁₆ C₂₄.₂₈D₄ D₇₂:₅C₂ D₆.₃D₁₂ D₂₄:₅S₃ D₁₂.₁₃D₆ D₁₂.₁₄D₆ C₂₄.₂₈D₆ D₁₂₀:₅C₂ D₂₄:₅D₅ Dic₁₀.₂₇D₆ D₁₂.D₁₀ D₁₂₀:₈C₂

Matrix representation of D₂₄:C₂ ►in GL₄(F₇) generated by

5	3	4	6
4	2	0	0
5	6	3	1
6	1	4	0

6	6	1	1
6	4	4	4
5	3	0	1
1	0	5	4

2	5	2	4
4	4	2	2
5	1	5	6
4	6	1	3

G:=sub<GL(4,GF(7))| [5,4,5,6,3,2,6,1,4,0,3,4,6,0,1,0],[6,6,5,1,6,4,3,0,1,4,0,5,1,4,1,4],[2,4,5,4,5,4,1,6,2,2,5,1,4,2,6,3] >;

D₂₄:C₂ in GAP, Magma, Sage, TeX

D_{24}\rtimes C_2

% in TeX

G:=Group("D24:C2");

// GroupNames label

G:=SmallGroup(96,126);

// by ID

G=gap.SmallGroup(96,126);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,362,116,86,297,159,69,2309]);

// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^17,c*b*c=a^4*b>;

// generators/relations

Export

Character table of D₂₄:C₂ in TeX

G = D24:C2 order 96 = 25·3

5th semidirect product of D24 and C2 acting faithfully

G = D₂₄:C₂ order 96 = 2⁵·3

5^th semidirect product of D₂₄ and C₂ acting faithfully