C24:9D4 - GroupNames

G = C₂₄:₉D₄ order 192 = 2⁶·3

9^th semidirect product of C₂₄ and D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₄:₉D₄, C₃:C₈:₁₀D₄, C₈:₆(C₃:D₄), C₃:₅(C₈:₃D₄), C₂₄:C₄:₆C₂, C₄.₂₆(S₃xD₄), (C₂xC₈).₉₁D₆, (C₂xD₂₄):₂₅C₂, C₁₂:₃D₄:₆C₂, (C₆xSD₁₆):₄C₂, (C₂xSD₁₆):₂S₃, (C₂xD₄).₇₇D₆, (C₂xQ₈).₈₂D₆, C₁₂.₁₇₉(C₂xD₄), C₁₂.₂₃D₄:₅C₂, C₆.₃₂(C₄:₁D₄), C₂.₃₁(Q₈:₃D₆), C₆.₈₁(C₈:C₂²), (C₂xDic₃).₇₄D₄, C₂².₂₇₃(S₃xD₄), (C₆xQ₈).₈₂C₂², C₂.₂₃(C₁₂:₃D₄), (C₂xC₁₂).₄₅₃C₂³, (C₂xC₂₄).₁₁₆C₂², (C₆xD₄).₁₀₂C₂², (C₂xD₁₂).₁₂₃C₂², (C₄xDic₃).₅₂C₂², (C₂xD₄:S₃):₂₁C₂, C₄.₁₀(C₂xC₃:D₄), (C₂xC₆).₃₆₅(C₂xD₄), (C₂xQ₈:₂S₃):₁₉C₂, (C₂xC₃:C₈).₁₆₁C₂², (C₂xC₄).₅₄₂(C₂²xS₃), SmallGroup(192,735)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — C₂₄:₉D₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂xC₁₂ — C₂xD₁₂ — C₂xD₂₄ — C₂₄:₉D₄

Lower central

C₃ — C₆ — C₂xC₁₂ — C₂₄:₉D₄

Upper central

C₁ — C₂² — C₂xC₄ — C₂xSD₁₆

Generators and relations for C₂₄:₉D₄
G = < a,b,c | a²⁴=b⁴=c²=1, bab^-1=a⁵, cac=a^-1, cbc=b^-1 >

Subgroups: 536 in 144 conjugacy classes, 43 normal (31 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂xC₈, C₂xC₈, D₈, SD₁₆, C₂xD₄, C₂xD₄, C₂xQ₈, C₃:C₈, C₂₄, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₈:C₄, C₄.₄D₄, C₄:₁D₄, C₂xD₈, C₂xSD₁₆, C₂xSD₁₆, D₂₄, C₂xC₃:C₈, C₄xDic₃, D₆:C₄, D₄:S₃, Q₈:₂S₃, C₂xC₂₄, C₃xSD₁₆, C₂xD₁₂, C₂xC₃:D₄, C₆xD₄, C₆xQ₈, C₈:₃D₄, C₂₄:C₄, C₂xD₂₄, C₂xD₄:S₃, C₁₂:₃D₄, C₂xQ₈:₂S₃, C₁₂.₂₃D₄, C₆xSD₁₆, C₂₄:₉D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₄:₁D₄, C₈:C₂², S₃xD₄, C₂xC₃:D₄, C₈:₃D₄, Q₈:₃D₆, C₁₂:₃D₄, C₂₄:₉D₄

Character table of C₂₄:₉D₄

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	8A	8B	8C	8D	12A	12B	12C	12D	24A	24B	24C	24D
size	1	1	1	1	8	24	24	2	2	2	8	12	12	2	2	2	8	8	4	4	12	12	4	4	8	8	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	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	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	-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	-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	-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	-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	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	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₉	2	-2	-2	2	0	0	0	2	-2	2	0	0	0	-2	-2	2	0	0	2	-2	0	0	2	-2	0	0	2	-2	-2	2	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	0	2	-2	-2	0	-2	2	2	2	2	0	0	0	0	0	0	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	0	0	-1	2	2	-2	0	0	-1	-1	-1	1	1	2	2	0	0	-1	-1	1	1	-1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₂	2	2	2	2	2	0	0	-1	2	2	-2	0	0	-1	-1	-1	-1	-1	-2	-2	0	0	-1	-1	1	1	1	1	1	1	orthogonal lifted from D₆
ρ₁₃	2	-2	-2	2	0	0	0	2	-2	2	0	0	0	-2	-2	2	0	0	-2	2	0	0	2	-2	0	0	-2	2	2	-2	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	0	0	-1	2	2	2	0	0	-1	-1	-1	-1	-1	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₅	2	-2	-2	2	0	0	0	2	2	-2	0	0	0	-2	-2	2	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₆	2	2	2	2	-2	0	0	-1	2	2	2	0	0	-1	-1	-1	1	1	-2	-2	0	0	-1	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₇	2	-2	-2	2	0	0	0	2	2	-2	0	0	0	-2	-2	2	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	2	2	0	0	0	2	-2	-2	0	2	-2	2	2	2	0	0	0	0	0	0	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	-2	2	0	0	0	-1	-2	2	0	0	0	1	1	-1	√-3	-√-3	2	-2	0	0	-1	1	-√-3	√-3	-1	1	1	-1	complex lifted from C₃:D₄
ρ₂₀	2	-2	-2	2	0	0	0	-1	-2	2	0	0	0	1	1	-1	√-3	-√-3	-2	2	0	0	-1	1	√-3	-√-3	1	-1	-1	1	complex lifted from C₃:D₄
ρ₂₁	2	-2	-2	2	0	0	0	-1	-2	2	0	0	0	1	1	-1	-√-3	√-3	-2	2	0	0	-1	1	-√-3	√-3	1	-1	-1	1	complex lifted from C₃:D₄
ρ₂₂	2	-2	-2	2	0	0	0	-1	-2	2	0	0	0	1	1	-1	-√-3	√-3	2	-2	0	0	-1	1	√-3	-√-3	-1	1	1	-1	complex lifted from C₃:D₄
ρ₂₃	4	-4	4	-4	0	0	0	4	0	0	0	0	0	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈:C₂²
ρ₂₄	4	4	4	4	0	0	0	-2	-4	-4	0	0	0	-2	-2	-2	0	0	0	0	0	0	2	2	0	0	0	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₅	4	-4	-4	4	0	0	0	-2	4	-4	0	0	0	2	2	-2	0	0	0	0	0	0	2	-2	0	0	0	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₆	4	4	-4	-4	0	0	0	4	0	0	0	0	0	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈:C₂²
ρ₂₇	4	4	-4	-4	0	0	0	-2	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	0	0	0	0	√6	√6	-√6	-√6	orthogonal lifted from Q₈:₃D₆
ρ₂₈	4	-4	4	-4	0	0	0	-2	0	0	0	0	0	2	-2	2	0	0	0	0	0	0	0	0	0	0	-√6	√6	-√6	√6	orthogonal lifted from Q₈:₃D₆
ρ₂₉	4	-4	4	-4	0	0	0	-2	0	0	0	0	0	2	-2	2	0	0	0	0	0	0	0	0	0	0	√6	-√6	√6	-√6	orthogonal lifted from Q₈:₃D₆
ρ₃₀	4	4	-4	-4	0	0	0	-2	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	0	0	0	0	-√6	-√6	√6	√6	orthogonal lifted from Q₈:₃D₆

Smallest permutation representation of C₂₄:₉D₄
►On 96 points

Generators in S₉₆

(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(1 68 47 86)(2 49 48 91)(3 54 25 96)(4 59 26 77)(5 64 27 82)(6 69 28 87)(7 50 29 92)(8 55 30 73)(9 60 31 78)(10 65 32 83)(11 70 33 88)(12 51 34 93)(13 56 35 74)(14 61 36 79)(15 66 37 84)(16 71 38 89)(17 52 39 94)(18 57 40 75)(19 62 41 80)(20 67 42 85)(21 72 43 90)(22 53 44 95)(23 58 45 76)(24 63 46 81)

(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(46 48)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 96)(59 95)(60 94)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,68,47,86)(2,49,48,91)(3,54,25,96)(4,59,26,77)(5,64,27,82)(6,69,28,87)(7,50,29,92)(8,55,30,73)(9,60,31,78)(10,65,32,83)(11,70,33,88)(12,51,34,93)(13,56,35,74)(14,61,36,79)(15,66,37,84)(16,71,38,89)(17,52,39,94)(18,57,40,75)(19,62,41,80)(20,67,42,85)(21,72,43,90)(22,53,44,95)(23,58,45,76)(24,63,46,81), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(46,48)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,68,47,86)(2,49,48,91)(3,54,25,96)(4,59,26,77)(5,64,27,82)(6,69,28,87)(7,50,29,92)(8,55,30,73)(9,60,31,78)(10,65,32,83)(11,70,33,88)(12,51,34,93)(13,56,35,74)(14,61,36,79)(15,66,37,84)(16,71,38,89)(17,52,39,94)(18,57,40,75)(19,62,41,80)(20,67,42,85)(21,72,43,90)(22,53,44,95)(23,58,45,76)(24,63,46,81), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(46,48)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82) );

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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,68,47,86),(2,49,48,91),(3,54,25,96),(4,59,26,77),(5,64,27,82),(6,69,28,87),(7,50,29,92),(8,55,30,73),(9,60,31,78),(10,65,32,83),(11,70,33,88),(12,51,34,93),(13,56,35,74),(14,61,36,79),(15,66,37,84),(16,71,38,89),(17,52,39,94),(18,57,40,75),(19,62,41,80),(20,67,42,85),(21,72,43,90),(22,53,44,95),(23,58,45,76),(24,63,46,81)], [(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(46,48),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,96),(59,95),(60,94),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82)]])

Matrix representation of C₂₄:₉D₄ ►in GL₆(F₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	34	39	34	39
0	0	34	68	34	68
0	0	39	34	34	39
0	0	39	5	34	68

17	70	0	0	0	0
48	56	0	0	0	0
0	0	69	65	7	0
0	0	69	4	66	66
0	0	66	0	69	65
0	0	7	7	69	4

1	0	0	0	0	0
60	72	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	72	0
0	0	0	0	1	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,39,39,0,0,39,68,34,5,0,0,34,34,34,34,0,0,39,68,39,68],[17,48,0,0,0,0,70,56,0,0,0,0,0,0,69,69,66,7,0,0,65,4,0,7,0,0,7,66,69,69,0,0,0,66,65,4],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;

C₂₄:₉D₄ in GAP, Magma, Sage, TeX

C_{24}\rtimes_9D_4

% in TeX

G:=Group("C24:9D4");

// GroupNames label

G:=SmallGroup(192,735);

// by ID

G=gap.SmallGroup(192,735);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,1684,438,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;

// generators/relations

Export

Character table of C₂₄:₉D₄ in TeX

G = C24:9D4 order 192 = 26·3

9th semidirect product of C24 and D4 acting via D4/C2=C22

G = C₂₄:₉D₄ order 192 = 2⁶·3

9^th semidirect product of C₂₄ and D₄ acting via D₄/C₂=C₂²