metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.100D6, C6.1002+ (1+4), (C4×D12)⋊12C2, Dic3⋊D4⋊5C2, C4⋊D12⋊4C2, C4⋊C4.275D6, C12⋊7D4⋊42C2, D6.D4⋊5C2, (C2×C6).79C24, C12.6Q8⋊5C2, C42⋊C2⋊19S3, C2.12(D4○D12), (C4×C12).30C22, D6⋊C4.64C22, C22⋊C4.103D6, (C22×C4).216D6, C12.237(C4○D4), C4.121(C4○D12), (C2×C12).152C23, (C2×D12).25C22, Dic3⋊C4.4C22, (C22×S3).27C23, C4⋊Dic3.294C22, (C22×C6).149C23, C22.108(S3×C23), C23.100(C22×S3), (C2×Dic3).32C23, (C22×C12).309C22, C3⋊1(C22.34C24), C6.35(C2×C4○D4), C2.38(C2×C4○D12), (S3×C2×C4).196C22, (C3×C42⋊C2)⋊21C2, (C3×C4⋊C4).315C22, (C2×C4).280(C22×S3), (C2×C3⋊D4).12C22, (C3×C22⋊C4).118C22, SmallGroup(192,1094)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 712 in 240 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×5], C3, C4 [×2], C4 [×9], C22, C22 [×15], S3 [×4], C6, C6 [×2], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], C23, C23 [×4], Dic3 [×4], C12 [×2], C12 [×5], D6 [×12], C2×C6, C2×C6 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4 [×10], C4×S3 [×4], D12 [×8], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×S3 [×4], C22×C6, C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×4], C2×D12 [×6], C2×C3⋊D4 [×4], C22×C12, C22.34C24, C12.6Q8, C4×D12 [×2], C4⋊D12, Dic3⋊D4 [×4], D6.D4 [×4], C12⋊7D4 [×2], C3×C42⋊C2, C42.100D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4) [×2], C4○D12 [×2], S3×C23, C22.34C24, C2×C4○D12, D4○D12 [×2], C42.100D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c5 >
►On 96 points
(1 90 7 96)(2 91 8 85)(3 92 9 86)(4 93 10 87)(5 94 11 88)(6 95 12 89)(13 56 19 50)(14 57 20 51)(15 58 21 52)(16 59 22 53)(17 60 23 54)(18 49 24 55)(25 67 31 61)(26 68 32 62)(27 69 33 63)(28 70 34 64)(29 71 35 65)(30 72 36 66)(37 82 43 76)(38 83 44 77)(39 84 45 78)(40 73 46 79)(41 74 47 80)(42 75 48 81)
(1 22 38 68)(2 17 39 63)(3 24 40 70)(4 19 41 65)(5 14 42 72)(6 21 43 67)(7 16 44 62)(8 23 45 69)(9 18 46 64)(10 13 47 71)(11 20 48 66)(12 15 37 61)(25 89 58 82)(26 96 59 77)(27 91 60 84)(28 86 49 79)(29 93 50 74)(30 88 51 81)(31 95 52 76)(32 90 53 83)(33 85 54 78)(34 92 55 73)(35 87 56 80)(36 94 57 75)
(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 89 44 76)(2 75 45 88)(3 87 46 74)(4 73 47 86)(5 85 48 84)(6 83 37 96)(7 95 38 82)(8 81 39 94)(9 93 40 80)(10 79 41 92)(11 91 42 78)(12 77 43 90)(13 34 65 49)(14 60 66 33)(15 32 67 59)(16 58 68 31)(17 30 69 57)(18 56 70 29)(19 28 71 55)(20 54 72 27)(21 26 61 53)(22 52 62 25)(23 36 63 51)(24 50 64 35)
G:=sub<Sym(96)| (1,90,7,96)(2,91,8,85)(3,92,9,86)(4,93,10,87)(5,94,11,88)(6,95,12,89)(13,56,19,50)(14,57,20,51)(15,58,21,52)(16,59,22,53)(17,60,23,54)(18,49,24,55)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(37,82,43,76)(38,83,44,77)(39,84,45,78)(40,73,46,79)(41,74,47,80)(42,75,48,81), (1,22,38,68)(2,17,39,63)(3,24,40,70)(4,19,41,65)(5,14,42,72)(6,21,43,67)(7,16,44,62)(8,23,45,69)(9,18,46,64)(10,13,47,71)(11,20,48,66)(12,15,37,61)(25,89,58,82)(26,96,59,77)(27,91,60,84)(28,86,49,79)(29,93,50,74)(30,88,51,81)(31,95,52,76)(32,90,53,83)(33,85,54,78)(34,92,55,73)(35,87,56,80)(36,94,57,75), (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), (1,89,44,76)(2,75,45,88)(3,87,46,74)(4,73,47,86)(5,85,48,84)(6,83,37,96)(7,95,38,82)(8,81,39,94)(9,93,40,80)(10,79,41,92)(11,91,42,78)(12,77,43,90)(13,34,65,49)(14,60,66,33)(15,32,67,59)(16,58,68,31)(17,30,69,57)(18,56,70,29)(19,28,71,55)(20,54,72,27)(21,26,61,53)(22,52,62,25)(23,36,63,51)(24,50,64,35)>;
G:=Group( (1,90,7,96)(2,91,8,85)(3,92,9,86)(4,93,10,87)(5,94,11,88)(6,95,12,89)(13,56,19,50)(14,57,20,51)(15,58,21,52)(16,59,22,53)(17,60,23,54)(18,49,24,55)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(37,82,43,76)(38,83,44,77)(39,84,45,78)(40,73,46,79)(41,74,47,80)(42,75,48,81), (1,22,38,68)(2,17,39,63)(3,24,40,70)(4,19,41,65)(5,14,42,72)(6,21,43,67)(7,16,44,62)(8,23,45,69)(9,18,46,64)(10,13,47,71)(11,20,48,66)(12,15,37,61)(25,89,58,82)(26,96,59,77)(27,91,60,84)(28,86,49,79)(29,93,50,74)(30,88,51,81)(31,95,52,76)(32,90,53,83)(33,85,54,78)(34,92,55,73)(35,87,56,80)(36,94,57,75), (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), (1,89,44,76)(2,75,45,88)(3,87,46,74)(4,73,47,86)(5,85,48,84)(6,83,37,96)(7,95,38,82)(8,81,39,94)(9,93,40,80)(10,79,41,92)(11,91,42,78)(12,77,43,90)(13,34,65,49)(14,60,66,33)(15,32,67,59)(16,58,68,31)(17,30,69,57)(18,56,70,29)(19,28,71,55)(20,54,72,27)(21,26,61,53)(22,52,62,25)(23,36,63,51)(24,50,64,35) );
G=PermutationGroup([(1,90,7,96),(2,91,8,85),(3,92,9,86),(4,93,10,87),(5,94,11,88),(6,95,12,89),(13,56,19,50),(14,57,20,51),(15,58,21,52),(16,59,22,53),(17,60,23,54),(18,49,24,55),(25,67,31,61),(26,68,32,62),(27,69,33,63),(28,70,34,64),(29,71,35,65),(30,72,36,66),(37,82,43,76),(38,83,44,77),(39,84,45,78),(40,73,46,79),(41,74,47,80),(42,75,48,81)], [(1,22,38,68),(2,17,39,63),(3,24,40,70),(4,19,41,65),(5,14,42,72),(6,21,43,67),(7,16,44,62),(8,23,45,69),(9,18,46,64),(10,13,47,71),(11,20,48,66),(12,15,37,61),(25,89,58,82),(26,96,59,77),(27,91,60,84),(28,86,49,79),(29,93,50,74),(30,88,51,81),(31,95,52,76),(32,90,53,83),(33,85,54,78),(34,92,55,73),(35,87,56,80),(36,94,57,75)], [(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)], [(1,89,44,76),(2,75,45,88),(3,87,46,74),(4,73,47,86),(5,85,48,84),(6,83,37,96),(7,95,38,82),(8,81,39,94),(9,93,40,80),(10,79,41,92),(11,91,42,78),(12,77,43,90),(13,34,65,49),(14,60,66,33),(15,32,67,59),(16,58,68,31),(17,30,69,57),(18,56,70,29),(19,28,71,55),(20,54,72,27),(21,26,61,53),(22,52,62,25),(23,36,63,51),(24,50,64,35)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 7 |
| 0 | 0 | 0 | 0 | 11 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 2 |
| 0 | 0 | 0 | 0 | 6 | 0 | 11 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 12 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 6 |
| 0 | 0 | 0 | 0 | 6 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 2 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,11,0,6,0,0,0,0,2,0,6,0,0,0,0,0,0,7,0,11,0,0,0,0,7,0,2,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[1,12,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,2,0,6,0,0,0,0,0,0,11,0,6,0,0,0,0,6,0,11,0,0,0,0,0,0,6,0,2] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | D4○D12 |
| kernel | C42.100D6 | C12.6Q8 | C4×D12 | C4⋊D12 | Dic3⋊D4 | D6.D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{100}D_6 % in TeX
G:=Group("C4^2.100D6"); // GroupNames label
G:=SmallGroup(192,1094);
// by ID
G=gap.SmallGroup(192,1094);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations