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## G = D12⋊17D4order 192 = 26·3

### 5th semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12⋊17D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C2×C4○D12 — D12⋊17D4
 Lower central C3 — C6 — C2×C12 — D12⋊17D4
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for D1217D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd=a6b, dcd=c-1 >

Subgroups: 496 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×10], S3 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8 [×3], C23, C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×6], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C22×C4, C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], C3⋊C8 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×S3, C22×C6, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C2×C3⋊C8 [×2], D4⋊S3 [×2], D4.S3 [×2], C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C2×C3⋊D4, C22×C12, C6×D4, C6×D4, D4⋊D4, C6.D8, C6.SD16, C12.55D4, C2×D4⋊S3, C2×D4.S3, C3×C4⋊D4, C2×C4○D12, D1217D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C4○D8, C8⋊C22, S3×D4 [×2], C2×C3⋊D4, D4⋊D4, D126C22, C232D6, Q8.13D6, D1217D4

Smallest permutation representation of D1217D4
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 21)(14 20)(15 19)(16 18)(22 24)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 39)(40 48)(41 47)(42 46)(43 45)(49 55)(50 54)(51 53)(56 60)(57 59)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 75)(76 84)(77 83)(78 82)(79 81)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)
(1 37 85 51)(2 44 86 58)(3 39 87 53)(4 46 88 60)(5 41 89 55)(6 48 90 50)(7 43 91 57)(8 38 92 52)(9 45 93 59)(10 40 94 54)(11 47 95 49)(12 42 96 56)(13 30 76 61)(14 25 77 68)(15 32 78 63)(16 27 79 70)(17 34 80 65)(18 29 81 72)(19 36 82 67)(20 31 83 62)(21 26 84 69)(22 33 73 64)(23 28 74 71)(24 35 75 66)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 92)(26 93)(27 94)(28 95)(29 96)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 73)(47 74)(48 75)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C4○D8 C8⋊C22 S3×D4 D12⋊6C22 Q8.13D6 kernel D12⋊17D4 C6.D8 C6.SD16 C12.55D4 C2×D4⋊S3 C2×D4.S3 C3×C4⋊D4 C2×C4○D12 C4⋊D4 Dic6 D12 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×D4 C2×C4 C23 C6 C6 C4 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 2 4 1 2 2 2

Matrix representation of D1217D4 in GL6(𝔽73)

 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 48 0 0 0 0 3 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 72 0 0 0 0 0 1 0 0 0 0 0 0 72 48 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 12 4 0 0 0 0 55 61 0 0 0 0 0 0 54 17 0 0 0 0 56 19
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 55 0 0 0 0 8 27 0 0 0 0 0 0 56 19 0 0 0 0 54 17

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,3,0,0,0,0,48,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,48,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,55,0,0,0,0,4,61,0,0,0,0,0,0,54,56,0,0,0,0,17,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,8,0,0,0,0,55,27,0,0,0,0,0,0,56,54,0,0,0,0,19,17] >;

D1217D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{17}D_4
% in TeX

G:=Group("D12:17D4");
// GroupNames label

G:=SmallGroup(192,596);
// by ID

G=gap.SmallGroup(192,596);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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