metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12)⋊17D4, C12⋊3D4⋊33C2, (C2×D4).237D6, C12.454(C2×D4), (C2×Q8).218D6, Dic3⋊4(C4○D4), (C2×C6).318C24, C6.168(C22×D4), (C22×C4).304D6, Dic3⋊Q8⋊34C2, C23.14D6⋊47C2, C12.23D4⋊34C2, C23.12D6⋊33C2, (C2×C12).889C23, D6⋊C4.160C22, (C6×D4).277C22, (C6×Q8).244C22, (C2×D12).282C22, (C22×C6).244C23, C23.149(C22×S3), C22.327(S3×C23), C3⋊7(C22.26C24), Dic3⋊C4.172C22, (C22×S3).139C23, (C22×C12).297C22, (C4×Dic3).261C22, (C2×Dic6).311C22, (C2×Dic3).294C23, C6.D4.137C22, (C22×Dic3).237C22, (C6×C4○D4)⋊10C2, (C2×C4○D4)⋊14S3, (C4×C3⋊D4)⋊61C2, (C2×C4×Dic3)⋊15C2, (C2×C6).83(C2×D4), (C2×C4○D12)⋊32C2, (C2×C4)⋊11(C3⋊D4), C6.218(C2×C4○D4), C2.106(S3×C4○D4), C4.146(C2×C3⋊D4), C22.1(C2×C3⋊D4), (S3×C2×C4).221C22, C2.41(C22×C3⋊D4), (C2×C4).832(C22×S3), (C2×C3⋊D4).142C22, SmallGroup(192,1391)
Series: Derived ►Chief ►Lower central ►Upper central
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], C4⋊1D4, 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], C12⋊3D4, 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
Generators and relations
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, dad=ab6, cbc-1=dbd=b5, dcd=c-1 >
►On 96 points
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 49)(23 50)(24 51)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 61)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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 52 82 45)(2 57 83 38)(3 50 84 43)(4 55 73 48)(5 60 74 41)(6 53 75 46)(7 58 76 39)(8 51 77 44)(9 56 78 37)(10 49 79 42)(11 54 80 47)(12 59 81 40)(13 94 70 30)(14 87 71 35)(15 92 72 28)(16 85 61 33)(17 90 62 26)(18 95 63 31)(19 88 64 36)(20 93 65 29)(21 86 66 34)(22 91 67 27)(23 96 68 32)(24 89 69 25)
(2 6)(3 11)(5 9)(8 12)(13 64)(14 69)(15 62)(16 67)(17 72)(18 65)(19 70)(20 63)(21 68)(22 61)(23 66)(24 71)(25 35)(26 28)(27 33)(29 31)(30 36)(32 34)(37 60)(38 53)(39 58)(40 51)(41 56)(42 49)(43 54)(44 59)(45 52)(46 57)(47 50)(48 55)(74 78)(75 83)(77 81)(80 84)(85 91)(86 96)(87 89)(88 94)(90 92)(93 95)
G:=sub<Sym(96)| (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,61)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,52,82,45)(2,57,83,38)(3,50,84,43)(4,55,73,48)(5,60,74,41)(6,53,75,46)(7,58,76,39)(8,51,77,44)(9,56,78,37)(10,49,79,42)(11,54,80,47)(12,59,81,40)(13,94,70,30)(14,87,71,35)(15,92,72,28)(16,85,61,33)(17,90,62,26)(18,95,63,31)(19,88,64,36)(20,93,65,29)(21,86,66,34)(22,91,67,27)(23,96,68,32)(24,89,69,25), (2,6)(3,11)(5,9)(8,12)(13,64)(14,69)(15,62)(16,67)(17,72)(18,65)(19,70)(20,63)(21,68)(22,61)(23,66)(24,71)(25,35)(26,28)(27,33)(29,31)(30,36)(32,34)(37,60)(38,53)(39,58)(40,51)(41,56)(42,49)(43,54)(44,59)(45,52)(46,57)(47,50)(48,55)(74,78)(75,83)(77,81)(80,84)(85,91)(86,96)(87,89)(88,94)(90,92)(93,95)>;
G:=Group( (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,61)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,52,82,45)(2,57,83,38)(3,50,84,43)(4,55,73,48)(5,60,74,41)(6,53,75,46)(7,58,76,39)(8,51,77,44)(9,56,78,37)(10,49,79,42)(11,54,80,47)(12,59,81,40)(13,94,70,30)(14,87,71,35)(15,92,72,28)(16,85,61,33)(17,90,62,26)(18,95,63,31)(19,88,64,36)(20,93,65,29)(21,86,66,34)(22,91,67,27)(23,96,68,32)(24,89,69,25), (2,6)(3,11)(5,9)(8,12)(13,64)(14,69)(15,62)(16,67)(17,72)(18,65)(19,70)(20,63)(21,68)(22,61)(23,66)(24,71)(25,35)(26,28)(27,33)(29,31)(30,36)(32,34)(37,60)(38,53)(39,58)(40,51)(41,56)(42,49)(43,54)(44,59)(45,52)(46,57)(47,50)(48,55)(74,78)(75,83)(77,81)(80,84)(85,91)(86,96)(87,89)(88,94)(90,92)(93,95) );
G=PermutationGroup([(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,49),(23,50),(24,51),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,61),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,52,82,45),(2,57,83,38),(3,50,84,43),(4,55,73,48),(5,60,74,41),(6,53,75,46),(7,58,76,39),(8,51,77,44),(9,56,78,37),(10,49,79,42),(11,54,80,47),(12,59,81,40),(13,94,70,30),(14,87,71,35),(15,92,72,28),(16,85,61,33),(17,90,62,26),(18,95,63,31),(19,88,64,36),(20,93,65,29),(21,86,66,34),(22,91,67,27),(23,96,68,32),(24,89,69,25)], [(2,6),(3,11),(5,9),(8,12),(13,64),(14,69),(15,62),(16,67),(17,72),(18,65),(19,70),(20,63),(21,68),(22,61),(23,66),(24,71),(25,35),(26,28),(27,33),(29,31),(30,36),(32,34),(37,60),(38,53),(39,58),(40,51),(41,56),(42,49),(43,54),(44,59),(45,52),(46,57),(47,50),(48,55),(74,78),(75,83),(77,81),(80,84),(85,91),(86,96),(87,89),(88,94),(90,92),(93,95)])
Matrix representation ►G ⊆ 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] >;
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 |
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