metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.98D6, C6.532- (1+4), C12⋊Q8⋊13C2, C4⋊C4.312D6, C42⋊3S3⋊6C2, C4.D12⋊13C2, (C4×Dic6)⋊11C2, (C2×C6).77C24, C42⋊C2⋊17S3, C4.98(C4○D12), (C4×C12).28C22, D6⋊C4.84C22, C2.11(Q8○D12), C22⋊C4.101D6, C23.8D6⋊4C2, (C22×C4).214D6, Dic6⋊C4⋊13C2, C12.200(C4○D4), C12.48D4⋊30C2, (C2×C12).698C23, Dic3⋊C4.3C22, C23.98(C22×S3), Dic3.34(C4○D4), (C22×S3).25C23, C4⋊Dic3.293C22, (C22×C6).147C23, C22.106(S3×C23), C23.11D6.1C2, (C2×Dic3).30C23, (C22×C12).234C22, C3⋊1(C22.50C24), (C4×Dic3).199C22, (C2×Dic6).232C22, C6.D4.99C22, C4⋊C4⋊7S3⋊13C2, C2.16(S3×C4○D4), C6.33(C2×C4○D4), (C4×C3⋊D4).6C2, C2.36(C2×C4○D12), (S3×C2×C4).62C22, (C3×C42⋊C2)⋊19C2, (C3×C4⋊C4).313C22, (C2×C4).279(C22×S3), (C2×C3⋊D4).106C22, (C3×C22⋊C4).116C22, SmallGroup(192,1092)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 472 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×13], C22, C22 [×6], S3, C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×5], D6 [×3], C2×C6, C2×C6 [×3], C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4, C22×C4, C2×D4, C2×Q8 [×3], Dic6 [×6], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×S3, C22×C6, C42⋊C2, C42⋊C2, C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3, C4×Dic3 [×4], Dic3⋊C4, Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×4], C6.D4, C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, C2×C3⋊D4, C22×C12, C22.50C24, C4×Dic6 [×2], C42⋊3S3 [×2], C23.8D6 [×2], C23.11D6 [×2], Dic6⋊C4, C12⋊Q8, C4⋊C4⋊7S3, C4.D12, C12.48D4, C4×C3⋊D4, C3×C42⋊C2, C42.98D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2- (1+4), C4○D12 [×2], S3×C23, C22.50C24, C2×C4○D12, S3×C4○D4, Q8○D12, C42.98D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 57 45 82)(2 52 46 77)(3 59 47 84)(4 54 48 79)(5 49 37 74)(6 56 38 81)(7 51 39 76)(8 58 40 83)(9 53 41 78)(10 60 42 73)(11 55 43 80)(12 50 44 75)(13 96 62 25)(14 91 63 32)(15 86 64 27)(16 93 65 34)(17 88 66 29)(18 95 67 36)(19 90 68 31)(20 85 69 26)(21 92 70 33)(22 87 71 28)(23 94 72 35)(24 89 61 30)
(1 61 7 67)(2 62 8 68)(3 63 9 69)(4 64 10 70)(5 65 11 71)(6 66 12 72)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)(25 58 31 52)(26 59 32 53)(27 60 33 54)(28 49 34 55)(29 50 35 56)(30 51 36 57)(73 92 79 86)(74 93 80 87)(75 94 81 88)(76 95 82 89)(77 96 83 90)(78 85 84 91)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 65 19 71)(14 70 20 64)(15 63 21 69)(16 68 22 62)(17 61 23 67)(18 66 24 72)(25 93 31 87)(26 86 32 92)(27 91 33 85)(28 96 34 90)(29 89 35 95)(30 94 36 88)(37 46 43 40)(38 39 44 45)(41 42 47 48)(49 52 55 58)(50 57 56 51)(53 60 59 54)(73 84 79 78)(74 77 80 83)(75 82 81 76)
G:=sub<Sym(96)| (1,57,45,82)(2,52,46,77)(3,59,47,84)(4,54,48,79)(5,49,37,74)(6,56,38,81)(7,51,39,76)(8,58,40,83)(9,53,41,78)(10,60,42,73)(11,55,43,80)(12,50,44,75)(13,96,62,25)(14,91,63,32)(15,86,64,27)(16,93,65,34)(17,88,66,29)(18,95,67,36)(19,90,68,31)(20,85,69,26)(21,92,70,33)(22,87,71,28)(23,94,72,35)(24,89,61,30), (1,61,7,67)(2,62,8,68)(3,63,9,69)(4,64,10,70)(5,65,11,71)(6,66,12,72)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39)(25,58,31,52)(26,59,32,53)(27,60,33,54)(28,49,34,55)(29,50,35,56)(30,51,36,57)(73,92,79,86)(74,93,80,87)(75,94,81,88)(76,95,82,89)(77,96,83,90)(78,85,84,91), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,65,19,71)(14,70,20,64)(15,63,21,69)(16,68,22,62)(17,61,23,67)(18,66,24,72)(25,93,31,87)(26,86,32,92)(27,91,33,85)(28,96,34,90)(29,89,35,95)(30,94,36,88)(37,46,43,40)(38,39,44,45)(41,42,47,48)(49,52,55,58)(50,57,56,51)(53,60,59,54)(73,84,79,78)(74,77,80,83)(75,82,81,76)>;
G:=Group( (1,57,45,82)(2,52,46,77)(3,59,47,84)(4,54,48,79)(5,49,37,74)(6,56,38,81)(7,51,39,76)(8,58,40,83)(9,53,41,78)(10,60,42,73)(11,55,43,80)(12,50,44,75)(13,96,62,25)(14,91,63,32)(15,86,64,27)(16,93,65,34)(17,88,66,29)(18,95,67,36)(19,90,68,31)(20,85,69,26)(21,92,70,33)(22,87,71,28)(23,94,72,35)(24,89,61,30), (1,61,7,67)(2,62,8,68)(3,63,9,69)(4,64,10,70)(5,65,11,71)(6,66,12,72)(13,40,19,46)(14,41,20,47)(15,42,21,48)(16,43,22,37)(17,44,23,38)(18,45,24,39)(25,58,31,52)(26,59,32,53)(27,60,33,54)(28,49,34,55)(29,50,35,56)(30,51,36,57)(73,92,79,86)(74,93,80,87)(75,94,81,88)(76,95,82,89)(77,96,83,90)(78,85,84,91), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,65,19,71)(14,70,20,64)(15,63,21,69)(16,68,22,62)(17,61,23,67)(18,66,24,72)(25,93,31,87)(26,86,32,92)(27,91,33,85)(28,96,34,90)(29,89,35,95)(30,94,36,88)(37,46,43,40)(38,39,44,45)(41,42,47,48)(49,52,55,58)(50,57,56,51)(53,60,59,54)(73,84,79,78)(74,77,80,83)(75,82,81,76) );
G=PermutationGroup([(1,57,45,82),(2,52,46,77),(3,59,47,84),(4,54,48,79),(5,49,37,74),(6,56,38,81),(7,51,39,76),(8,58,40,83),(9,53,41,78),(10,60,42,73),(11,55,43,80),(12,50,44,75),(13,96,62,25),(14,91,63,32),(15,86,64,27),(16,93,65,34),(17,88,66,29),(18,95,67,36),(19,90,68,31),(20,85,69,26),(21,92,70,33),(22,87,71,28),(23,94,72,35),(24,89,61,30)], [(1,61,7,67),(2,62,8,68),(3,63,9,69),(4,64,10,70),(5,65,11,71),(6,66,12,72),(13,40,19,46),(14,41,20,47),(15,42,21,48),(16,43,22,37),(17,44,23,38),(18,45,24,39),(25,58,31,52),(26,59,32,53),(27,60,33,54),(28,49,34,55),(29,50,35,56),(30,51,36,57),(73,92,79,86),(74,93,80,87),(75,94,81,88),(76,95,82,89),(77,96,83,90),(78,85,84,91)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,65,19,71),(14,70,20,64),(15,63,21,69),(16,68,22,62),(17,61,23,67),(18,66,24,72),(25,93,31,87),(26,86,32,92),(27,91,33,85),(28,96,34,90),(29,89,35,95),(30,94,36,88),(37,46,43,40),(38,39,44,45),(41,42,47,48),(49,52,55,58),(50,57,56,51),(53,60,59,54),(73,84,79,78),(74,77,80,83),(75,82,81,76)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 8 |
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 1 | 8 |
| 1 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,5,0,0,0,2,8],[11,4,0,0,9,2,0,0,0,0,8,0,0,0,0,8],[1,12,0,0,1,0,0,0,0,0,5,1,0,0,0,8],[1,0,0,0,1,12,0,0,0,0,5,0,0,0,0,5] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | ··· | 4S | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- (1+4) | S3×C4○D4 | Q8○D12 |
| kernel | C42.98D6 | C4×Dic6 | C42⋊3S3 | C23.8D6 | C23.11D6 | Dic6⋊C4 | C12⋊Q8 | C4⋊C4⋊7S3 | C4.D12 | C12.48D4 | C4×C3⋊D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{98}D_6 % in TeX
G:=Group("C4^2.98D6"); // GroupNames label
G:=SmallGroup(192,1092);
// by ID
G=gap.SmallGroup(192,1092);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,100,1571,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations