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## G = (C2×C12)⋊17D4order 192 = 26·3

### 13rd semidirect product of C2×C12 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C12)⋊17D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — (C2×C12)⋊17D4
 Lower central C3 — C2×C6 — (C2×C12)⋊17D4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for (C2×C12)⋊17D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, dad=ab6, cbc-1=dbd=b5, dcd=c-1 >

Subgroups: 744 in 310 conjugacy classes, 115 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×10], C22, C22 [×2], C22 [×14], S3 [×2], C6, C6 [×2], C6 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×20], Q8 [×4], C23, C23 [×2], C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8, C2×Q8, C4○D4 [×8], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×6], C2×C12 [×4], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×2], C22×C6, C22×C6 [×2], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, C4×Dic3 [×2], C4×Dic3 [×2], Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×6], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], C22.26C24, C2×C4×Dic3, C4×C3⋊D4 [×4], C23.12D6, C23.14D6 [×4], C123D4, Dic3⋊Q8, C12.23D4, C2×C4○D12, C6×C4○D4, (C2×C12)⋊17D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C2×C3⋊D4 [×6], S3×C23, C22.26C24, S3×C4○D4 [×2], C22×C3⋊D4, (C2×C12)⋊17D4

Smallest permutation representation of (C2×C12)⋊17D4
On 96 points
Generators in S96
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 33)(14 34)(15 35)(16 36)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(37 84)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(61 94)(62 95)(63 96)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(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 31 48 66)(2 36 37 71)(3 29 38 64)(4 34 39 69)(5 27 40 62)(6 32 41 67)(7 25 42 72)(8 30 43 65)(9 35 44 70)(10 28 45 63)(11 33 46 68)(12 26 47 61)(13 81 89 53)(14 74 90 58)(15 79 91 51)(16 84 92 56)(17 77 93 49)(18 82 94 54)(19 75 95 59)(20 80 96 52)(21 73 85 57)(22 78 86 50)(23 83 87 55)(24 76 88 60)
(2 6)(3 11)(5 9)(8 12)(13 91)(14 96)(15 89)(16 94)(17 87)(18 92)(19 85)(20 90)(21 95)(22 88)(23 93)(24 86)(25 72)(26 65)(27 70)(28 63)(29 68)(30 61)(31 66)(32 71)(33 64)(34 69)(35 62)(36 67)(37 41)(38 46)(40 44)(43 47)(49 55)(50 60)(51 53)(52 58)(54 56)(57 59)(73 75)(74 80)(76 78)(77 83)(79 81)(82 84)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 4 4 12 12 2 1 1 1 1 2 2 4 4 6 ··· 6 12 12 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 S3×C4○D4 kernel (C2×C12)⋊17D4 C2×C4×Dic3 C4×C3⋊D4 C23.12D6 C23.14D6 C12⋊3D4 Dic3⋊Q8 C12.23D4 C2×C4○D12 C6×C4○D4 C2×C4○D4 C2×C12 C22×C4 C2×D4 C2×Q8 Dic3 C2×C4 C2 # reps 1 1 4 1 4 1 1 1 1 1 1 4 3 3 1 8 8 4

Matrix representation of (C2×C12)⋊17D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 3 0 0 0 0 5 5
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 2 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,3,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,12] >;

(C2×C12)⋊17D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes_{17}D_4
% in TeX

G:=Group("(C2xC12):17D4");
// GroupNames label

G:=SmallGroup(192,1391);
// by ID

G=gap.SmallGroup(192,1391);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,297,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^6,c*b*c^-1=d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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