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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:C8:4D4, C4:D4:4S3, C3:3(C8:2D4), C4:C4.59D6, (C2xD4).39D6, C4.171(S3xD4), (C2xC12).72D4, C12:7D4:24C2, C6.D8:36C2, C12.148(C2xD4), (C22xC6).85D4, D4:Dic3:16C2, C6.94(C4:D4), C6.91(C8:C22), C12.Q8:35C2, (C6xD4).55C22, (C22xC4).137D6, C12.184(C4oD4), C2.13(D4:D6), C4.60(D4:2S3), (C2xC12).358C23, (C2xD12).98C22, C23.31(C3:D4), C2.12(D12:6C22), C4:Dic3.143C22, C2.15(C23.14D6), (C22xC12).162C22, (C2xD4:S3):11C2, (C3xC4:D4):4C2, (C2xC6).489(C2xD4), (C2xC4).50(C3:D4), (C2xC3:C8).109C22, (C2xC4.Dic3):11C2, (C3xC4:C4).106C22, (C2xC4).458(C22xS3), C22.164(C2xC3:D4), SmallGroup(192,598)

Series: Derived Chief Lower central Upper central

C1C2xC12 — C4:D4:S3
C1C3C6C12C2xC12C2xD12C12:7D4 — C4:D4:S3
C3C6C2xC12 — C4:D4:S3
C1C22C22xC4C4: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, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, M4(2), D8, C22xC4, C2xD4, C2xD4, C3:C8, C3:C8, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, D4:C4, C4.Q8, C4:D4, C4:D4, C2xM4(2), C2xD8, C2xC3:C8, C4.Dic3, C4:Dic3, D6:C4, D4:S3, C3xC22:C4, C3xC4:C4, C2xD12, C2xC3:D4, C22xC12, C6xD4, C6xD4, C8:2D4, C12.Q8, C6.D8, D4:Dic3, C2xC4.Dic3, C12:7D4, C2xD4:S3, C3xC4:D4, C4:D4:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, C8:C22, S3xD4, D4:2S3, C2xC3:D4, C8:2D4, D12:6C22, 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 C4oD4
ρ2222-2-200022-20002-2-20000-2i02i0-200200    complex lifted from C4oD4
ρ234-4-44000400000-4-4400000000000000    orthogonal lifted from C8:C22
ρ2444-4-4000-2-44000-22200000000-200200    orthogonal lifted from S3xD4
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 D4:2S3, Schur index 2
ρ294-4-44000-20000022-22-3-2-3000000000000    complex lifted from D12:6C22
ρ304-4-44000-20000022-2-2-32-3000000000000    complex lifted from D12:6C22

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

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