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## G = D4.13D12order 192 = 26·3

### 3rd non-split extension by D4 of D12 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D4.13D12
 Chief series C1 — C3 — C6 — C12 — D12 — C4○D12 — Q8○D12 — D4.13D12
 Lower central C3 — C6 — C12 — D4.13D12
 Upper central C1 — C2 — C4○D4 — C8○D4

Generators and relations for D4.13D12
G = < a,b,c,d | a4=b2=d2=1, c12=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c11 >

Subgroups: 624 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3 [×2], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×8], Q8, Q8 [×12], Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×8], C4○D4, C4○D4 [×12], C24, C24 [×3], Dic6 [×6], Dic6 [×6], C4×S3 [×6], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- 1+4 [×2], C24⋊C2 [×6], D24, Dic12 [×9], C2×C24 [×3], C3×M4(2) [×3], C2×Dic6 [×6], C4○D12 [×6], D42S3 [×6], S3×Q8 [×2], C3×C4○D4, Q8○D8, C4○D24 [×3], C2×Dic12 [×3], C8.D6 [×6], C3×C8○D4, Q8○D12 [×2], D4.13D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, Q8○D8, C22×D12, D4.13D12

Smallest permutation representation of D4.13D12
On 96 points
Generators in S96
(1 46 13 34)(2 47 14 35)(3 48 15 36)(4 25 16 37)(5 26 17 38)(6 27 18 39)(7 28 19 40)(8 29 20 41)(9 30 21 42)(10 31 22 43)(11 32 23 44)(12 33 24 45)(49 91 61 79)(50 92 62 80)(51 93 63 81)(52 94 64 82)(53 95 65 83)(54 96 66 84)(55 73 67 85)(56 74 68 86)(57 75 69 87)(58 76 70 88)(59 77 71 89)(60 78 72 90)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(43 48)(44 47)(45 46)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D12 D12 Q8○D8 D4.13D12 kernel D4.13D12 C4○D24 C2×Dic12 C8.D6 C3×C8○D4 Q8○D12 C8○D4 C3×D4 C3×Q8 C2×C8 M4(2) C4○D4 D4 Q8 C3 C1 # reps 1 3 3 6 1 2 1 3 1 3 3 1 6 2 2 4

Matrix representation of D4.13D12 in GL4(𝔽73) generated by

 0 0 1 0 0 0 0 1 72 0 0 0 0 72 0 0
,
 42 11 42 11 62 31 62 31 42 11 31 62 62 31 11 42
,
 50 55 0 0 18 68 0 0 0 0 50 55 0 0 18 68
,
 23 18 0 0 68 50 0 0 0 0 23 18 0 0 68 50
G:=sub<GL(4,GF(73))| [0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[42,62,42,62,11,31,11,31,42,62,31,11,11,31,62,42],[50,18,0,0,55,68,0,0,0,0,50,18,0,0,55,68],[23,68,0,0,18,50,0,0,0,0,23,68,0,0,18,50] >;

D4.13D12 in GAP, Magma, Sage, TeX

D_4._{13}D_{12}
% in TeX

G:=Group("D4.13D12");
// GroupNames label

G:=SmallGroup(192,1312);
// by ID

G=gap.SmallGroup(192,1312);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,192,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^12=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^11>;
// generators/relations

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