Copied to
clipboard

G = C4⋊D4⋊S3order 192 = 26·3

4th semidirect product of C4⋊D4 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C84D4, C4⋊D44S3, C33(C82D4), C4⋊C4.59D6, (C2×D4).39D6, C4.171(S3×D4), (C2×C12).72D4, C127D424C2, C6.D836C2, C12.148(C2×D4), (C22×C6).85D4, D4⋊Dic316C2, C6.94(C4⋊D4), C6.91(C8⋊C22), C12.Q835C2, (C6×D4).55C22, (C22×C4).137D6, C12.184(C4○D4), C2.13(D4⋊D6), C4.60(D42S3), (C2×C12).358C23, (C2×D12).98C22, C23.31(C3⋊D4), C2.12(D126C22), C4⋊Dic3.143C22, C2.15(C23.14D6), (C22×C12).162C22, (C2×D4⋊S3)⋊11C2, (C3×C4⋊D4)⋊4C2, (C2×C6).489(C2×D4), (C2×C4).50(C3⋊D4), (C2×C3⋊C8).109C22, (C2×C4.Dic3)⋊11C2, (C3×C4⋊C4).106C22, (C2×C4).458(C22×S3), C22.164(C2×C3⋊D4), SmallGroup(192,598)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C4⋊D4⋊S3
C1C3C6C12C2×C12C2×D12C127D4 — C4⋊D4⋊S3
C3C6C2×C12 — C4⋊D4⋊S3
C1C22C22×C4C4⋊D4

Generators and relations for C4⋊D4⋊S3
 G = < a,b,c,d,e | a4=b4=c2=d3=e2=1, bab-1=cac=eae=a-1, ad=da, cbc=b-1, bd=db, ebe=ab-1, cd=dc, ece=a-1c, ede=d-1 >

Subgroups: 416 in 130 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×9], S3, C6 [×3], C6 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], Dic3, C12 [×2], C12 [×2], D6 [×3], C2×C6, C2×C6 [×6], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×2], M4(2) [×2], D8 [×2], C22×C4, C2×D4, C2×D4 [×3], C3⋊C8 [×2], C3⋊C8, D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×S3, C22×C6, C22×C6, D4⋊C4 [×2], C4.Q8, C4⋊D4, C4⋊D4, C2×M4(2), C2×D8, C2×C3⋊C8 [×2], C4.Dic3 [×2], C4⋊Dic3, D6⋊C4, D4⋊S3 [×2], C3×C22⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C82D4, C12.Q8, C6.D8, D4⋊Dic3, C2×C4.Dic3, C127D4, C2×D4⋊S3, C3×C4⋊D4, C4⋊D4⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C8⋊C22 [×2], S3×D4, D42S3, C2×C3⋊D4, C82D4, D126C22, C23.14D6, D4⋊D6, C4⋊D4⋊S3

Character table of C4⋊D4⋊S3

 class 12A2B2C2D2E2F34A4B4C4D4E6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F
 size 111148242224824222448812121212444488
ρ1111111111111111111111111111111    trivial
ρ211111-111111-1111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ311111-1-11111-1-111111-1-111111111-1-1    linear of order 2
ρ4111111-111111-11111111-1-1-1-1111111    linear of order 2
ρ51111-111111-1-1-1111-1-111-11-111-1-11-1-1    linear of order 2
ρ61111-1-11111-11-1111-1-1-1-11-11-11-1-1111    linear of order 2
ρ71111-1-1-1111-111111-1-1-1-1-11-111-1-1111    linear of order 2
ρ81111-11-1111-1-11111-1-1111-11-11-1-11-1-1    linear of order 2
ρ922222-20-1222-20-1-1-1-1-1110000-1-1-1-111    orthogonal lifted from D6
ρ102222-220-122-2-20-1-1-111-1-10000-111-111    orthogonal lifted from D6
ρ1122-2-20002-220002-2-20000020-2200-200    orthogonal lifted from D4
ρ1222222002-2-2-20022222000000-2-2-2-200    orthogonal lifted from D4
ρ1322-2-20002-220002-2-200000-202200-200    orthogonal lifted from D4
ρ142222220-122220-1-1-1-1-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ152222-2-20-122-220-1-1-111110000-111-1-1-1    orthogonal lifted from D6
ρ162222-2002-2-2200222-2-2000000-222-200    orthogonal lifted from D4
ρ172222-200-1-2-2200-1-1-111--3-300001-1-11--3-3    complex lifted from C3⋊D4
ρ182222-200-1-2-2200-1-1-111-3--300001-1-11-3--3    complex lifted from C3⋊D4
ρ192222200-1-2-2-200-1-1-1-1-1--3-300001111-3--3    complex lifted from C3⋊D4
ρ202222200-1-2-2-200-1-1-1-1-1-3--300001111--3-3    complex lifted from C3⋊D4
ρ2122-2-200022-20002-2-200002i0-2i0-200200    complex lifted from C4○D4
ρ2222-2-200022-20002-2-20000-2i02i0-200200    complex lifted from C4○D4
ρ234-4-44000400000-4-4400000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-4000-2-44000-22200000000-200200    orthogonal lifted from S3×D4
ρ254-44-4000400000-44-400000000000000    orthogonal lifted from C8⋊C22
ρ264-44-4000-2000002-22000000000-2323000    orthogonal lifted from D4⋊D6
ρ274-44-4000-2000002-2200000000023-23000    orthogonal lifted from D4⋊D6
ρ2844-4-4000-24-4000-22200000000200-200    symplectic lifted from D42S3, Schur index 2
ρ294-4-44000-20000022-22-3-2-3000000000000    complex lifted from D126C22
ρ304-4-44000-20000022-2-2-32-3000000000000    complex lifted from D126C22

Smallest permutation representation of C4⋊D4⋊S3
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 82 19 80)(2 81 20 79)(3 84 17 78)(4 83 18 77)(5 87 26 15)(6 86 27 14)(7 85 28 13)(8 88 25 16)(9 89 23 93)(10 92 24 96)(11 91 21 95)(12 90 22 94)(29 66 35 58)(30 65 36 57)(31 68 33 60)(32 67 34 59)(37 55 49 76)(38 54 50 75)(39 53 51 74)(40 56 52 73)(41 62 45 71)(42 61 46 70)(43 64 47 69)(44 63 48 72)
(1 34)(2 33)(3 36)(4 35)(5 39)(6 38)(7 37)(8 40)(9 41)(10 44)(11 43)(12 42)(13 55)(14 54)(15 53)(16 56)(17 30)(18 29)(19 32)(20 31)(21 47)(22 46)(23 45)(24 48)(25 52)(26 51)(27 50)(28 49)(57 78)(58 77)(59 80)(60 79)(61 94)(62 93)(63 96)(64 95)(65 84)(66 83)(67 82)(68 81)(69 91)(70 90)(71 89)(72 92)(73 88)(74 87)(75 86)(76 85)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 78 95)(14 79 96)(15 80 93)(16 77 94)(17 21 28)(18 22 25)(19 23 26)(20 24 27)(29 46 52)(30 47 49)(31 48 50)(32 45 51)(33 44 38)(34 41 39)(35 42 40)(36 43 37)(53 59 62)(54 60 63)(55 57 64)(56 58 61)(65 69 76)(66 70 73)(67 71 74)(68 72 75)(81 92 86)(82 89 87)(83 90 88)(84 91 85)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 90)(14 89)(15 92)(16 91)(18 20)(21 28)(22 27)(23 26)(24 25)(29 32)(30 31)(33 36)(34 35)(37 44)(38 43)(39 42)(40 41)(45 52)(46 51)(47 50)(48 49)(53 69)(54 72)(55 71)(56 70)(57 67)(58 66)(59 65)(60 68)(61 73)(62 76)(63 75)(64 74)(77 84)(78 83)(79 82)(80 81)(85 94)(86 93)(87 96)(88 95)

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

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

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

Matrix representation of C4⋊D4⋊S3 in GL6(𝔽73)

100000
010000
000010
000001
0072000
0007200
,
47670000
52260000
002314959
005991423
009595059
0014231464
,
100000
40720000
004053568
006835540
0035683368
00540538
,
100000
010000
00727200
001000
00007272
000010
,
100000
40720000
001000
00727200
0000720
000011

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[47,52,0,0,0,0,67,26,0,0,0,0,0,0,23,59,9,14,0,0,14,9,59,23,0,0,9,14,50,14,0,0,59,23,59,64],[1,40,0,0,0,0,0,72,0,0,0,0,0,0,40,68,35,5,0,0,5,35,68,40,0,0,35,5,33,5,0,0,68,40,68,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,40,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;

C4⋊D4⋊S3 in GAP, Magma, Sage, TeX

C_4\rtimes D_4\rtimes S_3
% in TeX

G:=Group("C4:D4:S3");
// GroupNames label

G:=SmallGroup(192,598);
// by ID

G=gap.SmallGroup(192,598);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,555,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^3=e^2=1,b*a*b^-1=c*a*c=e*a*e=a^-1,a*d=d*a,c*b*c=b^-1,b*d=d*b,e*b*e=a*b^-1,c*d=d*c,e*c*e=a^-1*c,e*d*e=d^-1>;
// generators/relations

Export

Character table of C4⋊D4⋊S3 in TeX

׿
×
𝔽