metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.2Dic6, C4⋊C4.7D6, (C2×C8).7D6, C24⋊1C4⋊7C2, C3⋊2(D4.Q8), (C3×D4).2Q8, C12.4(C2×Q8), Dic3⋊C8⋊5C2, C4.4(C2×Dic6), D4⋊C4.3S3, (C2×D4).129D6, C6.21(C4○D8), C2.9(Q8⋊3D6), (C2×C24).7C22, C12.Q8⋊3C2, C4.Dic6⋊3C2, (D4×Dic3).6C2, C2.6(D8⋊3S3), C6.54(C8⋊C22), D4⋊Dic3.5C2, (C6×D4).27C22, C22.168(S3×D4), C6.10(C22⋊Q8), C12.148(C4○D4), C4.77(D4⋊2S3), (C2×C12).206C23, (C2×Dic3).138D4, C4⋊Dic3.65C22, (C4×Dic3).10C22, C2.15(Dic3.D4), (C2×C6).219(C2×D4), (C2×C3⋊C8).12C22, (C3×D4⋊C4).3C2, (C3×C4⋊C4).11C22, (C2×C4).313(C22×S3), SmallGroup(192,325)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.2Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=a2c6, bab=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd-1=c-1 >
Subgroups: 280 in 102 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, C22×Dic3, C6×D4, D4.Q8, C12.Q8, Dic3⋊C8, C24⋊1C4, D4⋊Dic3, C3×D4⋊C4, C4.Dic6, D4×Dic3, D4.2Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C4○D8, C8⋊C22, C2×Dic6, S3×D4, D4⋊2S3, D4.Q8, Dic3.D4, D8⋊3S3, Q8⋊3D6, D4.2Dic6
►On 96 points
(1 18 30 70)(2 71 31 19)(3 20 32 72)(4 61 33 21)(5 22 34 62)(6 63 35 23)(7 24 36 64)(8 65 25 13)(9 14 26 66)(10 67 27 15)(11 16 28 68)(12 69 29 17)(37 90 57 76)(38 77 58 91)(39 92 59 78)(40 79 60 93)(41 94 49 80)(42 81 50 95)(43 96 51 82)(44 83 52 85)(45 86 53 84)(46 73 54 87)(47 88 55 74)(48 75 56 89)
(1 24)(2 25)(3 14)(4 27)(5 16)(6 29)(7 18)(8 31)(9 20)(10 33)(11 22)(12 35)(13 19)(15 21)(17 23)(26 72)(28 62)(30 64)(32 66)(34 68)(36 70)(37 43)(38 85)(39 45)(40 87)(41 47)(42 89)(44 91)(46 93)(48 95)(49 55)(50 75)(51 57)(52 77)(53 59)(54 79)(56 81)(58 83)(60 73)(61 67)(63 69)(65 71)(74 94)(76 96)(78 86)(80 88)(82 90)(84 92)
(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 87 36 79)(2 86 25 78)(3 85 26 77)(4 96 27 76)(5 95 28 75)(6 94 29 74)(7 93 30 73)(8 92 31 84)(9 91 32 83)(10 90 33 82)(11 89 34 81)(12 88 35 80)(13 39 71 53)(14 38 72 52)(15 37 61 51)(16 48 62 50)(17 47 63 49)(18 46 64 60)(19 45 65 59)(20 44 66 58)(21 43 67 57)(22 42 68 56)(23 41 69 55)(24 40 70 54)
G:=sub<Sym(96)| (1,18,30,70)(2,71,31,19)(3,20,32,72)(4,61,33,21)(5,22,34,62)(6,63,35,23)(7,24,36,64)(8,65,25,13)(9,14,26,66)(10,67,27,15)(11,16,28,68)(12,69,29,17)(37,90,57,76)(38,77,58,91)(39,92,59,78)(40,79,60,93)(41,94,49,80)(42,81,50,95)(43,96,51,82)(44,83,52,85)(45,86,53,84)(46,73,54,87)(47,88,55,74)(48,75,56,89), (1,24)(2,25)(3,14)(4,27)(5,16)(6,29)(7,18)(8,31)(9,20)(10,33)(11,22)(12,35)(13,19)(15,21)(17,23)(26,72)(28,62)(30,64)(32,66)(34,68)(36,70)(37,43)(38,85)(39,45)(40,87)(41,47)(42,89)(44,91)(46,93)(48,95)(49,55)(50,75)(51,57)(52,77)(53,59)(54,79)(56,81)(58,83)(60,73)(61,67)(63,69)(65,71)(74,94)(76,96)(78,86)(80,88)(82,90)(84,92), (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), (1,87,36,79)(2,86,25,78)(3,85,26,77)(4,96,27,76)(5,95,28,75)(6,94,29,74)(7,93,30,73)(8,92,31,84)(9,91,32,83)(10,90,33,82)(11,89,34,81)(12,88,35,80)(13,39,71,53)(14,38,72,52)(15,37,61,51)(16,48,62,50)(17,47,63,49)(18,46,64,60)(19,45,65,59)(20,44,66,58)(21,43,67,57)(22,42,68,56)(23,41,69,55)(24,40,70,54)>;
G:=Group( (1,18,30,70)(2,71,31,19)(3,20,32,72)(4,61,33,21)(5,22,34,62)(6,63,35,23)(7,24,36,64)(8,65,25,13)(9,14,26,66)(10,67,27,15)(11,16,28,68)(12,69,29,17)(37,90,57,76)(38,77,58,91)(39,92,59,78)(40,79,60,93)(41,94,49,80)(42,81,50,95)(43,96,51,82)(44,83,52,85)(45,86,53,84)(46,73,54,87)(47,88,55,74)(48,75,56,89), (1,24)(2,25)(3,14)(4,27)(5,16)(6,29)(7,18)(8,31)(9,20)(10,33)(11,22)(12,35)(13,19)(15,21)(17,23)(26,72)(28,62)(30,64)(32,66)(34,68)(36,70)(37,43)(38,85)(39,45)(40,87)(41,47)(42,89)(44,91)(46,93)(48,95)(49,55)(50,75)(51,57)(52,77)(53,59)(54,79)(56,81)(58,83)(60,73)(61,67)(63,69)(65,71)(74,94)(76,96)(78,86)(80,88)(82,90)(84,92), (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), (1,87,36,79)(2,86,25,78)(3,85,26,77)(4,96,27,76)(5,95,28,75)(6,94,29,74)(7,93,30,73)(8,92,31,84)(9,91,32,83)(10,90,33,82)(11,89,34,81)(12,88,35,80)(13,39,71,53)(14,38,72,52)(15,37,61,51)(16,48,62,50)(17,47,63,49)(18,46,64,60)(19,45,65,59)(20,44,66,58)(21,43,67,57)(22,42,68,56)(23,41,69,55)(24,40,70,54) );
G=PermutationGroup([[(1,18,30,70),(2,71,31,19),(3,20,32,72),(4,61,33,21),(5,22,34,62),(6,63,35,23),(7,24,36,64),(8,65,25,13),(9,14,26,66),(10,67,27,15),(11,16,28,68),(12,69,29,17),(37,90,57,76),(38,77,58,91),(39,92,59,78),(40,79,60,93),(41,94,49,80),(42,81,50,95),(43,96,51,82),(44,83,52,85),(45,86,53,84),(46,73,54,87),(47,88,55,74),(48,75,56,89)], [(1,24),(2,25),(3,14),(4,27),(5,16),(6,29),(7,18),(8,31),(9,20),(10,33),(11,22),(12,35),(13,19),(15,21),(17,23),(26,72),(28,62),(30,64),(32,66),(34,68),(36,70),(37,43),(38,85),(39,45),(40,87),(41,47),(42,89),(44,91),(46,93),(48,95),(49,55),(50,75),(51,57),(52,77),(53,59),(54,79),(56,81),(58,83),(60,73),(61,67),(63,69),(65,71),(74,94),(76,96),(78,86),(80,88),(82,90),(84,92)], [(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)], [(1,87,36,79),(2,86,25,78),(3,85,26,77),(4,96,27,76),(5,95,28,75),(6,94,29,74),(7,93,30,73),(8,92,31,84),(9,91,32,83),(10,90,33,82),(11,89,34,81),(12,88,35,80),(13,39,71,53),(14,38,72,52),(15,37,61,51),(16,48,62,50),(17,47,63,49),(18,46,64,60),(19,45,65,59),(20,44,66,58),(21,43,67,57),(22,42,68,56),(23,41,69,55),(24,40,70,54)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 6 | 6 | 8 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | C4○D4 | Dic6 | C4○D8 | C8⋊C22 | D4⋊2S3 | S3×D4 | D8⋊3S3 | Q8⋊3D6 |
| kernel | D4.2Dic6 | C12.Q8 | Dic3⋊C8 | C24⋊1C4 | D4⋊Dic3 | C3×D4⋊C4 | C4.Dic6 | D4×Dic3 | D4⋊C4 | C2×Dic3 | C3×D4 | C4⋊C4 | C2×C8 | C2×D4 | C12 | D4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D4.2Dic6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 48 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 70 |
| 0 | 0 | 0 | 1 |
| 59 | 66 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 32 | 48 |
| 0 | 0 | 38 | 41 |
| 59 | 5 | 0 | 0 |
| 19 | 14 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,48,0,0,3,72],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,70,1],[59,7,0,0,66,66,0,0,0,0,32,38,0,0,48,41],[59,19,0,0,5,14,0,0,0,0,27,0,0,0,0,27] >;
D4.2Dic6 in GAP, Magma, Sage, TeX
D_4._2{\rm Dic}_6 % in TeX
G:=Group("D4.2Dic6"); // GroupNames label
G:=SmallGroup(192,325);
// by ID
G=gap.SmallGroup(192,325);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,926,219,226,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=a^2*c^6,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations