metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊5Dic6, C42.103D6, C6.132+ (1+4), (C3×D4)⋊6Q8, C12⋊Q8⋊15C2, C4⋊C4.278D6, (C4×D4).11S3, C3⋊2(D4⋊3Q8), C12.42(C2×Q8), (C4×Dic6)⋊26C2, (D4×C12).12C2, (C2×D4).242D6, (C2×C6).83C24, C4.15(C2×Dic6), C12.48D4⋊7C2, C6.13(C22×Q8), C22⋊C4.106D6, C12.3Q8⋊14C2, C12.6Q8⋊14C2, (D4×Dic3).11C2, (C22×C4).218D6, C2.16(D4⋊6D6), C22.1(C2×Dic6), (C2×C12).154C23, (C4×C12).146C22, Dic3.D4⋊7C2, (C6×D4).249C22, C4⋊Dic3.37C22, C2.15(C22×Dic6), Dic3.19(C4○D4), (C22×C12).77C22, (C22×C6).153C23, C22.111(S3×C23), C23.173(C22×S3), (C2×Dic3).33C23, (C2×Dic6).25C22, (C4×Dic3).72C22, C6.D4.8C22, Dic3⋊C4.108C22, (C22×Dic3).91C22, (C2×C6).3(C2×Q8), C2.18(S3×C4○D4), C6.137(C2×C4○D4), (C2×Dic3⋊C4)⋊24C2, (C3×C4⋊C4).319C22, (C2×C4).154(C22×S3), (C3×C22⋊C4).104C22, SmallGroup(192,1098)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 504 in 228 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×13], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×2], Dic3 [×7], C12 [×2], C12 [×4], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×3], Dic6 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×C6 [×2], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×3], C4⋊Dic3 [×2], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C2×Dic6 [×2], C22×Dic3 [×4], C22×C12 [×2], C6×D4, D4⋊3Q8, C4×Dic6, C12.6Q8, Dic3.D4 [×4], C12⋊Q8, C12.3Q8, C2×Dic3⋊C4 [×2], C12.48D4 [×2], D4×Dic3 [×2], D4×C12, D4⋊5Dic6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C2×Dic6 [×6], S3×C23, D4⋊3Q8, C22×Dic6, D4⋊6D6, S3×C4○D4, D4⋊5Dic6
Generators and relations
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >
►On 96 points
(1 36 82 19)(2 20 83 25)(3 26 84 21)(4 22 73 27)(5 28 74 23)(6 24 75 29)(7 30 76 13)(8 14 77 31)(9 32 78 15)(10 16 79 33)(11 34 80 17)(12 18 81 35)(37 85 68 59)(38 60 69 86)(39 87 70 49)(40 50 71 88)(41 89 72 51)(42 52 61 90)(43 91 62 53)(44 54 63 92)(45 93 64 55)(46 56 65 94)(47 95 66 57)(48 58 67 96)
(1 30)(2 14)(3 32)(4 16)(5 34)(6 18)(7 36)(8 20)(9 26)(10 22)(11 28)(12 24)(13 82)(15 84)(17 74)(19 76)(21 78)(23 80)(25 77)(27 79)(29 81)(31 83)(33 73)(35 75)(37 53)(38 92)(39 55)(40 94)(41 57)(42 96)(43 59)(44 86)(45 49)(46 88)(47 51)(48 90)(50 65)(52 67)(54 69)(56 71)(58 61)(60 63)(62 85)(64 87)(66 89)(68 91)(70 93)(72 95)
(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 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 96 19 90)(14 95 20 89)(15 94 21 88)(16 93 22 87)(17 92 23 86)(18 91 24 85)(25 51 31 57)(26 50 32 56)(27 49 33 55)(28 60 34 54)(29 59 35 53)(30 58 36 52)(61 76 67 82)(62 75 68 81)(63 74 69 80)(64 73 70 79)(65 84 71 78)(66 83 72 77)
G:=sub<Sym(96)| (1,36,82,19)(2,20,83,25)(3,26,84,21)(4,22,73,27)(5,28,74,23)(6,24,75,29)(7,30,76,13)(8,14,77,31)(9,32,78,15)(10,16,79,33)(11,34,80,17)(12,18,81,35)(37,85,68,59)(38,60,69,86)(39,87,70,49)(40,50,71,88)(41,89,72,51)(42,52,61,90)(43,91,62,53)(44,54,63,92)(45,93,64,55)(46,56,65,94)(47,95,66,57)(48,58,67,96), (1,30)(2,14)(3,32)(4,16)(5,34)(6,18)(7,36)(8,20)(9,26)(10,22)(11,28)(12,24)(13,82)(15,84)(17,74)(19,76)(21,78)(23,80)(25,77)(27,79)(29,81)(31,83)(33,73)(35,75)(37,53)(38,92)(39,55)(40,94)(41,57)(42,96)(43,59)(44,86)(45,49)(46,88)(47,51)(48,90)(50,65)(52,67)(54,69)(56,71)(58,61)(60,63)(62,85)(64,87)(66,89)(68,91)(70,93)(72,95), (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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,96,19,90)(14,95,20,89)(15,94,21,88)(16,93,22,87)(17,92,23,86)(18,91,24,85)(25,51,31,57)(26,50,32,56)(27,49,33,55)(28,60,34,54)(29,59,35,53)(30,58,36,52)(61,76,67,82)(62,75,68,81)(63,74,69,80)(64,73,70,79)(65,84,71,78)(66,83,72,77)>;
G:=Group( (1,36,82,19)(2,20,83,25)(3,26,84,21)(4,22,73,27)(5,28,74,23)(6,24,75,29)(7,30,76,13)(8,14,77,31)(9,32,78,15)(10,16,79,33)(11,34,80,17)(12,18,81,35)(37,85,68,59)(38,60,69,86)(39,87,70,49)(40,50,71,88)(41,89,72,51)(42,52,61,90)(43,91,62,53)(44,54,63,92)(45,93,64,55)(46,56,65,94)(47,95,66,57)(48,58,67,96), (1,30)(2,14)(3,32)(4,16)(5,34)(6,18)(7,36)(8,20)(9,26)(10,22)(11,28)(12,24)(13,82)(15,84)(17,74)(19,76)(21,78)(23,80)(25,77)(27,79)(29,81)(31,83)(33,73)(35,75)(37,53)(38,92)(39,55)(40,94)(41,57)(42,96)(43,59)(44,86)(45,49)(46,88)(47,51)(48,90)(50,65)(52,67)(54,69)(56,71)(58,61)(60,63)(62,85)(64,87)(66,89)(68,91)(70,93)(72,95), (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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,96,19,90)(14,95,20,89)(15,94,21,88)(16,93,22,87)(17,92,23,86)(18,91,24,85)(25,51,31,57)(26,50,32,56)(27,49,33,55)(28,60,34,54)(29,59,35,53)(30,58,36,52)(61,76,67,82)(62,75,68,81)(63,74,69,80)(64,73,70,79)(65,84,71,78)(66,83,72,77) );
G=PermutationGroup([(1,36,82,19),(2,20,83,25),(3,26,84,21),(4,22,73,27),(5,28,74,23),(6,24,75,29),(7,30,76,13),(8,14,77,31),(9,32,78,15),(10,16,79,33),(11,34,80,17),(12,18,81,35),(37,85,68,59),(38,60,69,86),(39,87,70,49),(40,50,71,88),(41,89,72,51),(42,52,61,90),(43,91,62,53),(44,54,63,92),(45,93,64,55),(46,56,65,94),(47,95,66,57),(48,58,67,96)], [(1,30),(2,14),(3,32),(4,16),(5,34),(6,18),(7,36),(8,20),(9,26),(10,22),(11,28),(12,24),(13,82),(15,84),(17,74),(19,76),(21,78),(23,80),(25,77),(27,79),(29,81),(31,83),(33,73),(35,75),(37,53),(38,92),(39,55),(40,94),(41,57),(42,96),(43,59),(44,86),(45,49),(46,88),(47,51),(48,90),(50,65),(52,67),(54,69),(56,71),(58,61),(60,63),(62,85),(64,87),(66,89),(68,91),(70,93),(72,95)], [(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)], [(1,42,7,48),(2,41,8,47),(3,40,9,46),(4,39,10,45),(5,38,11,44),(6,37,12,43),(13,96,19,90),(14,95,20,89),(15,94,21,88),(16,93,22,87),(17,92,23,86),(18,91,24,85),(25,51,31,57),(26,50,32,56),(27,49,33,55),(28,60,34,54),(29,59,35,53),(30,58,36,52),(61,76,67,82),(62,75,68,81),(63,74,69,80),(64,73,70,79),(65,84,71,78),(66,83,72,77)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,3,10,0,0,0,0,7,10,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | D6 | C4○D4 | Dic6 | 2+ (1+4) | D4⋊6D6 | S3×C4○D4 |
| kernel | D4⋊5Dic6 | C4×Dic6 | C12.6Q8 | Dic3.D4 | C12⋊Q8 | C12.3Q8 | C2×Dic3⋊C4 | C12.48D4 | D4×Dic3 | D4×C12 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | D4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_4\rtimes_5Dic_6
% in TeX
G:=Group("D4:5Dic6"); // GroupNames label
G:=SmallGroup(192,1098);
// by ID
G=gap.SmallGroup(192,1098);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,675,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations