D24:9C4 - GroupNames

G = D₂₄:₉C₄ order 192 = 2⁶·3

9^th semidirect product of D₂₄ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₄:₉C₄, C₈:₄(C₄xS₃), C₂₄:₈(C₂xC₄), C₄.Q₈:₅S₃, D₁₂:₆(C₂xC₄), C₂₄:C₄:₃C₂, (C₂xC₈).₆₃D₆, C₆.₅₁(C₄xD₄), C₃:₃(D₈:C₄), C₄:C₄.₁₆₃D₆, Dic₃:₅D₄:₆C₂, (C₂xD₂₄).₁₃C₂, C₆.D₈:₁₈C₂, C₂.₆(Q₈:₃D₆), C₂².₈₆(S₃xD₄), C₁₂.₃₄(C₄oD₄), C₆.₇₂(C₈:C₂²), C₁₂.₄₅(C₂²xC₄), C₄.₆(Q₈:₃S₃), (C₂xC₂₄).₁₁₂C₂², (C₂xC₁₂).₂₈₅C₂³, (C₂xDic₃).₁₆₅D₄, (C₂xD₁₂).₇₇C₂², C₂.₁₁(Dic₃:₅D₄), (C₄xDic₃).₃₁C₂², C₄.₄₂(S₃xC₂xC₄), (C₃xC₄.Q₈):₅C₂, (C₂xC₆).₂₉₀(C₂xD₄), (C₂xC₃:C₈).₆₂C₂², (C₃xC₄:C₄).₇₈C₂², (C₂xC₄).₃₈₈(C₂²xS₃), SmallGroup(192,428)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₂₄:₉C₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂xC₁₂ — C₂xD₁₂ — C₂xD₂₄ — D₂₄:₉C₄

Lower central

C₃ — C₆ — C₁₂ — D₂₄:₉C₄

Upper central

C₁ — C₂² — C₂xC₄ — C₄.Q₈

Generators and relations for D₂₄:₉C₄
G = < a,b,c | a²⁴=b²=c⁴=1, bab=a^-1, cac^-1=a¹⁹, cbc^-1=a¹⁸b >

Subgroups: 448 in 132 conjugacy classes, 49 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, D₄, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, C₂xC₈, D₈, C₂²xC₄, C₂xD₄, C₃:C₈, C₂₄, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₈:C₄, D₄:C₄, C₄.Q₈, C₄xD₄, C₂xD₈, D₂₄, C₂xC₃:C₈, C₄xDic₃, D₆:C₄, C₃xC₄:C₄, C₂xC₂₄, S₃xC₂xC₄, C₂xD₁₂, D₈:C₄, C₆.D₈, C₂₄:C₄, C₃xC₄.Q₈, Dic₃:₅D₄, C₂xD₂₄, D₂₄:₉C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, D₆, C₂²xC₄, C₂xD₄, C₄oD₄, C₄xS₃, C₂²xS₃, C₄xD₄, C₈:C₂², S₃xC₂xC₄, S₃xD₄, Q₈:₃S₃, D₈:C₄, Dic₃:₅D₄, Q₈:₃D₆, D₂₄:₉C₄

Smallest permutation representation of D₂₄:₉C₄
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(68 72)(69 71)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	6A	6B	6C	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	24A	24B	24C	24D
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	4	4	6	6	6	8	8	8	8	12	12	12	12	12	12	24	24	24	24
size	1	1	1	1	12	12	12	12	2	2	2	4	4	4	4	6	6	6	6	2	2	2	4	4	12	12	4	4	8	8	8	8	4	4	4	4

36 irreducible representations

dim

type

image

C₁

C₂

C₄

S₃

D₄

D₆

C₄oD₄

C₄xS₃

C₈:C₂²

Q₈:₃S₃

S₃xD₄

Q₈:₃D₆

kernel

D₂₄:₉C₄

C₆.D₈

C₂₄:C₄

C₃xC₄.Q₈

Dic₃:₅D₄

C₂xD₂₄

D₂₄

C₄.Q₈

C₂xDic₃

C₄:C₄

C₂xC₈

C₁₂

C₈

C₆

C₄

C₂²

C₂

# reps

Matrix representation of D₂₄:₉C₄ ►in GL₈(F₇₃)

48	48	54	35	0	0	0	0
27	19	19	54	0	0	0	0
56	38	54	25	0	0	0	0
36	56	46	25	0	0	0	0
0	0	0	0	0	0	0	5
0	0	0	0	0	0	68	0
0	0	0	0	29	0	5	5
0	0	0	0	29	29	68	10

1	0	0	0	0	0	0	0
1	72	0	0	0	0	0	0
9	55	0	1	0	0	0	0
9	55	1	0	0	0	0	0
0	0	0	0	0	1	50	48
0	0	0	0	1	0	48	48
0	0	0	0	0	0	1	0
0	0	0	0	0	0	1	72

0	0	72	1	0	0	0	0
27	27	71	72	0	0	0	0
0	0	46	0	0	0	0	0
72	0	46	0	0	0	0	0
0	0	0	0	30	60	0	13
0	0	0	0	13	43	60	0
0	0	0	0	0	17	30	13
0	0	0	0	56	0	60	43

G:=sub<GL(8,GF(73))| [48,27,56,36,0,0,0,0,48,19,38,56,0,0,0,0,54,19,54,46,0,0,0,0,35,54,25,25,0,0,0,0,0,0,0,0,0,0,29,29,0,0,0,0,0,0,0,29,0,0,0,0,0,68,5,68,0,0,0,0,5,0,5,10],[1,1,9,9,0,0,0,0,0,72,55,55,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,50,48,1,1,0,0,0,0,48,48,0,72],[0,27,0,72,0,0,0,0,0,27,0,0,0,0,0,0,72,71,46,46,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,30,13,0,56,0,0,0,0,60,43,17,0,0,0,0,0,0,60,30,60,0,0,0,0,13,0,13,43] >;

D₂₄:₉C₄ in GAP, Magma, Sage, TeX

D_{24}\rtimes_9C_4

% in TeX

G:=Group("D24:9C4");

// GroupNames label

G:=SmallGroup(192,428);

// by ID

G=gap.SmallGroup(192,428);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,219,58,1684,438,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;

// generators/relations

48	48	54	35	0	0	0	0
27	19	19	54	0	0	0	0
56	38	54	25	0	0	0	0
36	56	46	25	0	0	0	0
0	0	0	0	0	0	0	5
0	0	0	0	0	0	68	0
0	0	0	0	29	0	5	5
0	0	0	0	29	29	68	10

1	0	0	0	0	0	0	0
1	72	0	0	0	0	0	0
9	55	0	1	0	0	0	0
9	55	1	0	0	0	0	0
0	0	0	0	0	1	50	48
0	0	0	0	1	0	48	48
0	0	0	0	0	0	1	0
0	0	0	0	0	0	1	72

0	0	72	1	0	0	0	0
27	27	71	72	0	0	0	0
0	0	46	0	0	0	0	0
72	0	46	0	0	0	0	0
0	0	0	0	30	60	0	13
0	0	0	0	13	43	60	0
0	0	0	0	0	17	30	13
0	0	0	0	56	0	60	43

48	48	54	35	0	0	0	0
27	19	19	54	0	0	0	0
56	38	54	25	0	0	0	0
36	56	46	25	0	0	0	0
0	0	0	0	0	0	0	5
0	0	0	0	0	0	68	0
0	0	0	0	29	0	5	5
0	0	0	0	29	29	68	10

1	0	0	0	0	0	0	0
1	72	0	0	0	0	0	0
9	55	0	1	0	0	0	0
9	55	1	0	0	0	0	0
0	0	0	0	0	1	50	48
0	0	0	0	1	0	48	48
0	0	0	0	0	0	1	0
0	0	0	0	0	0	1	72

0	0	72	1	0	0	0	0
27	27	71	72	0	0	0	0
0	0	46	0	0	0	0	0
72	0	46	0	0	0	0	0
0	0	0	0	30	60	0	13
0	0	0	0	13	43	60	0
0	0	0	0	0	17	30	13
0	0	0	0	56	0	60	43

G = D24:9C4 order 192 = 26·3

9th semidirect product of D24 and C4 acting via C4/C2=C2

G = D₂₄:₉C₄ order 192 = 2⁶·3

9^th semidirect product of D₂₄ and C₄ acting via C₄/C₂=C₂

48	48	54	35	0	0	0	0
27	19	19	54	0	0	0	0
56	38	54	25	0	0	0	0
36	56	46	25	0	0	0	0
0	0	0	0	0	0	0	5
0	0	0	0	0	0	68	0
0	0	0	0	29	0	5	5
0	0	0	0	29	29	68	10

1	0	0	0	0	0	0	0
1	72	0	0	0	0	0	0
9	55	0	1	0	0	0	0
9	55	1	0	0	0	0	0
0	0	0	0	0	1	50	48
0	0	0	0	1	0	48	48
0	0	0	0	0	0	1	0
0	0	0	0	0	0	1	72

0	0	72	1	0	0	0	0
27	27	71	72	0	0	0	0
0	0	46	0	0	0	0	0
72	0	46	0	0	0	0	0
0	0	0	0	30	60	0	13
0	0	0	0	13	43	60	0
0	0	0	0	0	17	30	13
0	0	0	0	56	0	60	43