C24:7D4 - GroupNames

G = C₂₄:₇D₄ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₄:₇D₄, C₈:₂D₁₂, C₄.Q₈:₃S₃, C₃:₂(C₈:₂D₄), C₄:C₄.₃₉D₆, (C₂xC₈).₆₁D₆, (C₂xD₂₄):₂₄C₂, C₁₂:D₄:₆C₂, C₄.₅₁(C₂xD₁₂), C₁₂.₁₃₁(C₂xD₄), C₆.D₈:₁₆C₂, C₁₂.₃₀(C₄oD₄), C₂.₂₂(Q₈:₃D₆), C₆.₄₄(C₄:D₄), C₆.₇₀(C₈:C₂²), C₄.₄(Q₈:₃S₃), (C₂xDic₃).₄₂D₄, (C₂²xS₃).₂₄D₄, C₂².₂₁₇(S₃xD₄), C₂.₁₇(C₁₂:D₄), (C₂xC₁₂).₂₈₁C₂³, (C₂xC₂₄).₁₁₀C₂², (C₂xD₁₂).₇₅C₂², (C₃xC₄.Q₈):₃C₂, (C₂xC₈:S₃):₂C₂, (C₂xC₆).₂₈₆(C₂xD₄), (C₂xC₃:C₈).₅₈C₂², (S₃xC₂xC₄).₃₃C₂², (C₃xC₄:C₄).₇₄C₂², (C₂xC₄).₃₈₄(C₂²xS₃), SmallGroup(192,424)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — C₂₄:₇D₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂xC₁₂ — S₃xC₂xC₄ — C₂xC₈:S₃ — C₂₄:₇D₄

Lower central

C₃ — C₆ — C₂xC₁₂ — C₂₄:₇D₄

Upper central

C₁ — C₂² — C₂xC₄ — C₄.Q₈

Generators and relations for C₂₄:₇D₄
G = < a,b,c | a²⁴=b⁴=c²=1, bab^-1=a¹⁹, cac=a^-1, cbc=b^-1 >

Subgroups: 512 in 130 conjugacy classes, 41 normal (23 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, D₄, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₂xC₈, C₂xC₈, M₄(2), D₈, C₂²xC₄, C₂xD₄, C₃:C₈, C₂₄, C₄xS₃, D₁₂, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²xS₃, D₄:C₄, C₄.Q₈, C₄:D₄, C₂xM₄(2), C₂xD₈, C₈:S₃, D₂₄, C₂xC₃:C₈, D₆:C₄, C₃xC₄:C₄, C₂xC₂₄, S₃xC₂xC₄, C₂xD₁₂, C₂xD₁₂, C₈:₂D₄, C₆.D₈, C₃xC₄.Q₈, C₁₂:D₄, C₂xC₈:S₃, C₂xD₂₄, C₂₄:₇D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₄oD₄, D₁₂, C₂²xS₃, C₄:D₄, C₈:C₂², C₂xD₁₂, S₃xD₄, Q₈:₃S₃, C₈:₂D₄, C₁₂:D₄, Q₈:₃D₆, C₂₄:₇D₄

Character table of C₂₄:₇D₄

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	4D	4E	6A	6B	6C	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	24A	24B	24C	24D
size	1	1	1	1	12	24	24	2	2	2	8	8	12	2	2	2	4	4	12	12	4	4	8	8	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	-1	2	2	-2	2	0	-1	-1	-1	-2	-2	0	0	-1	-1	1	-1	-1	1	1	1	1	1	orthogonal lifted from D₆
ρ₁₀	2	2	2	2	2	0	0	2	-2	-2	0	0	-2	2	2	2	0	0	0	0	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	0	-1	2	2	2	2	0	-1	-1	-1	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₂	2	2	2	2	0	0	0	-1	2	2	2	-2	0	-1	-1	-1	-2	-2	0	0	-1	-1	-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	2	-2	0	0	-2	2	0	0	0	0	-2	2	-2	2	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	0	0	2	-2	-2	0	0	2	2	2	2	0	0	0	0	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	2	0	0	0	-1	2	2	-2	-2	0	-1	-1	-1	2	2	0	0	-1	-1	1	1	1	1	-1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₆	2	2	-2	-2	0	0	0	2	-2	2	0	0	0	-2	2	-2	-2	2	0	0	-2	2	0	0	0	0	2	-2	2	-2	orthogonal lifted from D₄
ρ₁₇	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	2	-2	0	0	1	-1	√3	-√3	√3	-√3	1	-1	1	-1	orthogonal lifted from D₁₂
ρ₁₈	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	2	-2	0	0	1	-1	-√3	√3	-√3	√3	1	-1	1	-1	orthogonal lifted from D₁₂
ρ₁₉	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	-2	2	0	0	1	-1	√3	√3	-√3	-√3	-1	1	-1	1	orthogonal lifted from D₁₂
ρ₂₀	2	2	-2	-2	0	0	0	-1	-2	2	0	0	0	1	-1	1	-2	2	0	0	1	-1	-√3	-√3	√3	√3	-1	1	-1	1	orthogonal lifted from D₁₂
ρ₂₁	2	2	-2	-2	0	0	0	2	2	-2	0	0	0	-2	2	-2	0	0	2i	-2i	2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₂	2	2	-2	-2	0	0	0	2	2	-2	0	0	0	-2	2	-2	0	0	-2i	2i	2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄oD₄
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	-2	2	0	0	0	0	0	0	0	0	orthogonal lifted from Q₈:₃S₃, Schur index 2
ρ₂₅	4	4	4	4	0	0	0	-2	-4	-4	0	0	0	-2	-2	-2	0	0	0	0	2	2	0	0	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 C₂₄:₇D₄
►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 74 40 70)(2 93 41 65)(3 88 42 60)(4 83 43 55)(5 78 44 50)(6 73 45 69)(7 92 46 64)(8 87 47 59)(9 82 48 54)(10 77 25 49)(11 96 26 68)(12 91 27 63)(13 86 28 58)(14 81 29 53)(15 76 30 72)(16 95 31 67)(17 90 32 62)(18 85 33 57)(19 80 34 52)(20 75 35 71)(21 94 36 66)(22 89 37 61)(23 84 38 56)(24 79 39 51)

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

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

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

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

Matrix representation of C₂₄:₇D₄ ►in GL₈(F₇₃)

55	18	32	32	0	0	0	0
55	37	41	0	0	0	0	0
60	60	18	55	0	0	0	0
13	0	18	36	0	0	0	0
0	0	0	0	1	1	11	20
0	0	0	0	60	1	0	20
0	0	0	0	32	32	59	21
0	0	0	0	56	59	40	12

7	14	71	0	0	0	0	0
59	66	0	71	0	0	0	0
0	0	66	59	0	0	0	0
0	0	14	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	21	21	72	3
0	0	0	0	1	0	0	0
0	0	0	0	48	48	0	52

72	0	0	0	0	0	0	0
1	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	72	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	1	0
0	0	0	0	59	0	25	72

G:=sub<GL(8,GF(73))| [55,55,60,13,0,0,0,0,18,37,60,0,0,0,0,0,32,41,18,18,0,0,0,0,32,0,55,36,0,0,0,0,0,0,0,0,1,60,32,56,0,0,0,0,1,1,32,59,0,0,0,0,11,0,59,40,0,0,0,0,20,20,21,12],[7,59,0,0,0,0,0,0,14,66,0,0,0,0,0,0,71,0,66,14,0,0,0,0,0,71,59,7,0,0,0,0,0,0,0,0,0,21,1,48,0,0,0,0,0,21,0,48,0,0,0,0,1,72,0,0,0,0,0,0,0,3,0,52],[72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,59,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,0,0,72] >;

C₂₄:₇D₄ in GAP, Magma, Sage, TeX

C_{24}\rtimes_7D_4

% in TeX

G:=Group("C24:7D4");

// GroupNames label

G:=SmallGroup(192,424);

// by ID

G=gap.SmallGroup(192,424);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,254,555,58,438,102,6278]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂₄:₇D₄ in TeX

55	18	32	32	0	0	0	0
55	37	41	0	0	0	0	0
60	60	18	55	0	0	0	0
13	0	18	36	0	0	0	0
0	0	0	0	1	1	11	20
0	0	0	0	60	1	0	20
0	0	0	0	32	32	59	21
0	0	0	0	56	59	40	12

7	14	71	0	0	0	0	0
59	66	0	71	0	0	0	0
0	0	66	59	0	0	0	0
0	0	14	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	21	21	72	3
0	0	0	0	1	0	0	0
0	0	0	0	48	48	0	52

72	0	0	0	0	0	0	0
1	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	72	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	1	0
0	0	0	0	59	0	25	72

55	18	32	32	0	0	0	0
55	37	41	0	0	0	0	0
60	60	18	55	0	0	0	0
13	0	18	36	0	0	0	0
0	0	0	0	1	1	11	20
0	0	0	0	60	1	0	20
0	0	0	0	32	32	59	21
0	0	0	0	56	59	40	12

7	14	71	0	0	0	0	0
59	66	0	71	0	0	0	0
0	0	66	59	0	0	0	0
0	0	14	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	21	21	72	3
0	0	0	0	1	0	0	0
0	0	0	0	48	48	0	52

72	0	0	0	0	0	0	0
1	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	72	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	1	0
0	0	0	0	59	0	25	72

G = C24:7D4 order 192 = 26·3

7th semidirect product of C24 and D4 acting via D4/C2=C22

G = C₂₄:₇D₄ order 192 = 2⁶·3

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

55	18	32	32	0	0	0	0
55	37	41	0	0	0	0	0
60	60	18	55	0	0	0	0
13	0	18	36	0	0	0	0
0	0	0	0	1	1	11	20
0	0	0	0	60	1	0	20
0	0	0	0	32	32	59	21
0	0	0	0	56	59	40	12

7	14	71	0	0	0	0	0
59	66	0	71	0	0	0	0
0	0	66	59	0	0	0	0
0	0	14	7	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	21	21	72	3
0	0	0	0	1	0	0	0
0	0	0	0	48	48	0	52

72	0	0	0	0	0	0	0
1	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	72	72	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	72	0	0
0	0	0	0	0	0	1	0
0	0	0	0	59	0	25	72