metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4)⋊9D12, C6.2(C4×D4), (C2×D12)⋊5C4, (C2×C12)⋊19D4, C2.5(C4×D12), C6.2C22≀C2, D6⋊1(C22⋊C4), (C2×Dic3)⋊15D4, C6.3(C4⋊D4), (C22×C4).31D6, C22.59(S3×D4), C2.2(Dic3⋊D4), C2.1(D6⋊D4), C2.1(C12⋊D4), C2.C42⋊8S3, (C22×D12).1C2, (C22×S3).66D4, C22.24(C2×D12), C2.5(Dic3⋊5D4), C6.C42⋊27C2, C2.3(D6.D4), C3⋊1(C23.23D4), C22.34(C4○D12), (S3×C23).82C22, C23.265(C22×S3), (C22×C6).290C23, (C22×C12).331C22, C22.17(Q8⋊3S3), C6.38(C22.D4), (C22×Dic3).14C22, (C2×C4)⋊2(C4×S3), (C2×C12)⋊4(C2×C4), (C2×D6⋊C4)⋊28C2, (S3×C22×C4)⋊12C2, C2.7(S3×C22⋊C4), C6.4(C2×C22⋊C4), C22.89(S3×C2×C4), (C22×S3)⋊3(C2×C4), (C2×C6).199(C2×D4), (C2×C6).49(C22×C4), (C2×C6).183(C4○D4), (C3×C2.C42)⋊14C2, SmallGroup(192,224)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊9D12
G = < a,b,c,d | a2=b4=c12=d2=1, cbc-1=dbd=ab=ba, ac=ca, ad=da, dcd=c-1 >
Subgroups: 896 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, D6⋊C4, S3×C2×C4, C2×D12, C2×D12, C22×Dic3, C22×C12, S3×C23, C23.23D4, C6.C42, C3×C2.C42, C2×D6⋊C4, S3×C22×C4, C22×D12, (C2×C4)⋊9D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, S3×C2×C4, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C23.23D4, C4×D12, S3×C22⋊C4, D6⋊D4, Dic3⋊D4, Dic3⋊5D4, D6.D4, C12⋊D4, (C2×C4)⋊9D12
►On 96 points
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 73)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 85)(22 86)(23 87)(24 88)(25 59)(26 60)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 68)(38 69)(39 70)(40 71)(41 72)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)
(1 92 39 54)(2 17 40 33)(3 94 41 56)(4 19 42 35)(5 96 43 58)(6 21 44 25)(7 86 45 60)(8 23 46 27)(9 88 47 50)(10 13 48 29)(11 90 37 52)(12 15 38 31)(14 68 30 84)(16 70 32 74)(18 72 34 76)(20 62 36 78)(22 64 26 80)(24 66 28 82)(49 81 87 65)(51 83 89 67)(53 73 91 69)(55 75 93 71)(57 77 95 61)(59 79 85 63)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 85)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 90)(21 89)(22 88)(23 87)(24 86)(25 51)(26 50)(27 49)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 43)(38 42)(39 41)(44 48)(45 47)(61 69)(62 68)(63 67)(64 66)(70 72)(73 77)(74 76)(78 84)(79 83)(80 82)
G:=sub<Sym(96)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,73)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(25,59)(26,60)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,68)(38,69)(39,70)(40,71)(41,72)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67), (1,92,39,54)(2,17,40,33)(3,94,41,56)(4,19,42,35)(5,96,43,58)(6,21,44,25)(7,86,45,60)(8,23,46,27)(9,88,47,50)(10,13,48,29)(11,90,37,52)(12,15,38,31)(14,68,30,84)(16,70,32,74)(18,72,34,76)(20,62,36,78)(22,64,26,80)(24,66,28,82)(49,81,87,65)(51,83,89,67)(53,73,91,69)(55,75,93,71)(57,77,95,61)(59,79,85,63), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,85)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,90)(21,89)(22,88)(23,87)(24,86)(25,51)(26,50)(27,49)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,43)(38,42)(39,41)(44,48)(45,47)(61,69)(62,68)(63,67)(64,66)(70,72)(73,77)(74,76)(78,84)(79,83)(80,82)>;
G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,73)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(25,59)(26,60)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,68)(38,69)(39,70)(40,71)(41,72)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67), (1,92,39,54)(2,17,40,33)(3,94,41,56)(4,19,42,35)(5,96,43,58)(6,21,44,25)(7,86,45,60)(8,23,46,27)(9,88,47,50)(10,13,48,29)(11,90,37,52)(12,15,38,31)(14,68,30,84)(16,70,32,74)(18,72,34,76)(20,62,36,78)(22,64,26,80)(24,66,28,82)(49,81,87,65)(51,83,89,67)(53,73,91,69)(55,75,93,71)(57,77,95,61)(59,79,85,63), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,85)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,90)(21,89)(22,88)(23,87)(24,86)(25,51)(26,50)(27,49)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,43)(38,42)(39,41)(44,48)(45,47)(61,69)(62,68)(63,67)(64,66)(70,72)(73,77)(74,76)(78,84)(79,83)(80,82) );
G=PermutationGroup([[(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,73),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,85),(22,86),(23,87),(24,88),(25,59),(26,60),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,68),(38,69),(39,70),(40,71),(41,72),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67)], [(1,92,39,54),(2,17,40,33),(3,94,41,56),(4,19,42,35),(5,96,43,58),(6,21,44,25),(7,86,45,60),(8,23,46,27),(9,88,47,50),(10,13,48,29),(11,90,37,52),(12,15,38,31),(14,68,30,84),(16,70,32,74),(18,72,34,76),(20,62,36,78),(22,64,26,80),(24,66,28,82),(49,81,87,65),(51,83,89,67),(53,73,91,69),(55,75,93,71),(57,77,95,61),(59,79,85,63)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,85),(14,96),(15,95),(16,94),(17,93),(18,92),(19,91),(20,90),(21,89),(22,88),(23,87),(24,86),(25,51),(26,50),(27,49),(28,60),(29,59),(30,58),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,43),(38,42),(39,41),(44,48),(45,47),(61,69),(62,68),(63,67),(64,66),(70,72),(73,77),(74,76),(78,84),(79,83),(80,82)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 | S3×D4 | Q8⋊3S3 |
| kernel | (C2×C4)⋊9D12 | C6.C42 | C3×C2.C42 | C2×D6⋊C4 | S3×C22×C4 | C22×D12 | C2×D12 | C2.C42 | C2×Dic3 | C2×C12 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 3 | 1 | 1 | 8 | 1 | 2 | 2 | 4 | 3 | 4 | 4 | 4 | 4 | 3 | 1 |
Matrix representation of (C2×C4)⋊9D12 ►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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 3 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,1,0,0,0,0,2,8,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,10,10,0,0,0,0,3,7],[1,8,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12] >;
(C2×C4)⋊9D12 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_9D_{12} % in TeX
G:=Group("(C2xC4):9D12"); // GroupNames label
G:=SmallGroup(192,224);
// by ID
G=gap.SmallGroup(192,224);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=d^2=1,c*b*c^-1=d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,d*c*d=c^-1>;
// generators/relations