metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.178D6, (D4×Dic3)⋊16C2, (C2×D4).152D6, C4⋊D4.10S3, C22⋊C4.47D6, (C2×C6).144C24, C12.3Q8⋊18C2, (C22×C4).383D6, Dic6⋊C4⋊21C2, C12.201(C4○D4), C23.12D6⋊15C2, C4.67(D4⋊2S3), C12.48D4⋊31C2, (C2×C12).501C23, (C6×D4).118C22, C23.16D6⋊4C2, C23.8D6⋊14C2, C23.23D6⋊7C2, C23.21(C22×S3), (C22×C6).15C23, Dic3.41(C4○D4), C22.5(D4⋊2S3), Dic3⋊C4.15C22, C4⋊Dic3.205C22, C22.165(S3×C23), (C4×Dic3).91C22, (C22×C12).238C22, C3⋊6(C23.36C23), (C2×Dic6).152C22, (C2×Dic3).226C23, C6.D4.21C22, (C22×Dic3).105C22, (C2×C4×Dic3)⋊8C2, C2.35(S3×C4○D4), C6.149(C2×C4○D4), (C3×C4⋊D4).7C2, (C2×C6).20(C4○D4), C2.32(C2×D4⋊2S3), (C3×C4⋊C4).140C22, (C2×C4).292(C22×S3), (C3×C22⋊C4).9C22, SmallGroup(192,1159)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 496 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×2], Dic3 [×7], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×Dic3 [×4], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×C6, C22×C6 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×4], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×2], C6.D4 [×8], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C6×D4, C6×D4 [×2], C23.36C23, C23.16D6 [×2], C23.8D6 [×2], Dic6⋊C4, C12.3Q8, C2×C4×Dic3, C12.48D4, D4×Dic3, D4×Dic3 [×2], C23.23D6 [×2], C23.12D6, C3×C4⋊D4, C4⋊C4.178D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], D4⋊2S3 [×4], S3×C23, C23.36C23, C2×D4⋊2S3 [×2], S3×C4○D4, C4⋊C4.178D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >
►On 96 points
(1 16 19 26)(2 17 20 27)(3 18 21 28)(4 13 22 29)(5 14 23 30)(6 15 24 25)(7 45 33 68)(8 46 34 69)(9 47 35 70)(10 48 36 71)(11 43 31 72)(12 44 32 67)(37 58 95 89)(38 59 96 90)(39 60 91 85)(40 55 92 86)(41 56 93 87)(42 57 94 88)(49 61 84 78)(50 62 79 73)(51 63 80 74)(52 64 81 75)(53 65 82 76)(54 66 83 77)
(1 69 72 4)(2 5 67 70)(3 71 68 6)(7 25 18 36)(8 31 13 26)(9 27 14 32)(10 33 15 28)(11 29 16 34)(12 35 17 30)(19 46 43 22)(20 23 44 47)(21 48 45 24)(37 73 76 40)(38 41 77 74)(39 75 78 42)(49 88 60 81)(50 82 55 89)(51 90 56 83)(52 84 57 85)(53 86 58 79)(54 80 59 87)(61 94 91 64)(62 65 92 95)(63 96 93 66)
(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 94 43 75)(2 93 44 74)(3 92 45 73)(4 91 46 78)(5 96 47 77)(6 95 48 76)(7 79 28 55)(8 84 29 60)(9 83 30 59)(10 82 25 58)(11 81 26 57)(12 80 27 56)(13 85 34 49)(14 90 35 54)(15 89 36 53)(16 88 31 52)(17 87 32 51)(18 86 33 50)(19 42 72 64)(20 41 67 63)(21 40 68 62)(22 39 69 61)(23 38 70 66)(24 37 71 65)
G:=sub<Sym(96)| (1,16,19,26)(2,17,20,27)(3,18,21,28)(4,13,22,29)(5,14,23,30)(6,15,24,25)(7,45,33,68)(8,46,34,69)(9,47,35,70)(10,48,36,71)(11,43,31,72)(12,44,32,67)(37,58,95,89)(38,59,96,90)(39,60,91,85)(40,55,92,86)(41,56,93,87)(42,57,94,88)(49,61,84,78)(50,62,79,73)(51,63,80,74)(52,64,81,75)(53,65,82,76)(54,66,83,77), (1,69,72,4)(2,5,67,70)(3,71,68,6)(7,25,18,36)(8,31,13,26)(9,27,14,32)(10,33,15,28)(11,29,16,34)(12,35,17,30)(19,46,43,22)(20,23,44,47)(21,48,45,24)(37,73,76,40)(38,41,77,74)(39,75,78,42)(49,88,60,81)(50,82,55,89)(51,90,56,83)(52,84,57,85)(53,86,58,79)(54,80,59,87)(61,94,91,64)(62,65,92,95)(63,96,93,66), (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), (1,94,43,75)(2,93,44,74)(3,92,45,73)(4,91,46,78)(5,96,47,77)(6,95,48,76)(7,79,28,55)(8,84,29,60)(9,83,30,59)(10,82,25,58)(11,81,26,57)(12,80,27,56)(13,85,34,49)(14,90,35,54)(15,89,36,53)(16,88,31,52)(17,87,32,51)(18,86,33,50)(19,42,72,64)(20,41,67,63)(21,40,68,62)(22,39,69,61)(23,38,70,66)(24,37,71,65)>;
G:=Group( (1,16,19,26)(2,17,20,27)(3,18,21,28)(4,13,22,29)(5,14,23,30)(6,15,24,25)(7,45,33,68)(8,46,34,69)(9,47,35,70)(10,48,36,71)(11,43,31,72)(12,44,32,67)(37,58,95,89)(38,59,96,90)(39,60,91,85)(40,55,92,86)(41,56,93,87)(42,57,94,88)(49,61,84,78)(50,62,79,73)(51,63,80,74)(52,64,81,75)(53,65,82,76)(54,66,83,77), (1,69,72,4)(2,5,67,70)(3,71,68,6)(7,25,18,36)(8,31,13,26)(9,27,14,32)(10,33,15,28)(11,29,16,34)(12,35,17,30)(19,46,43,22)(20,23,44,47)(21,48,45,24)(37,73,76,40)(38,41,77,74)(39,75,78,42)(49,88,60,81)(50,82,55,89)(51,90,56,83)(52,84,57,85)(53,86,58,79)(54,80,59,87)(61,94,91,64)(62,65,92,95)(63,96,93,66), (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), (1,94,43,75)(2,93,44,74)(3,92,45,73)(4,91,46,78)(5,96,47,77)(6,95,48,76)(7,79,28,55)(8,84,29,60)(9,83,30,59)(10,82,25,58)(11,81,26,57)(12,80,27,56)(13,85,34,49)(14,90,35,54)(15,89,36,53)(16,88,31,52)(17,87,32,51)(18,86,33,50)(19,42,72,64)(20,41,67,63)(21,40,68,62)(22,39,69,61)(23,38,70,66)(24,37,71,65) );
G=PermutationGroup([(1,16,19,26),(2,17,20,27),(3,18,21,28),(4,13,22,29),(5,14,23,30),(6,15,24,25),(7,45,33,68),(8,46,34,69),(9,47,35,70),(10,48,36,71),(11,43,31,72),(12,44,32,67),(37,58,95,89),(38,59,96,90),(39,60,91,85),(40,55,92,86),(41,56,93,87),(42,57,94,88),(49,61,84,78),(50,62,79,73),(51,63,80,74),(52,64,81,75),(53,65,82,76),(54,66,83,77)], [(1,69,72,4),(2,5,67,70),(3,71,68,6),(7,25,18,36),(8,31,13,26),(9,27,14,32),(10,33,15,28),(11,29,16,34),(12,35,17,30),(19,46,43,22),(20,23,44,47),(21,48,45,24),(37,73,76,40),(38,41,77,74),(39,75,78,42),(49,88,60,81),(50,82,55,89),(51,90,56,83),(52,84,57,85),(53,86,58,79),(54,80,59,87),(61,94,91,64),(62,65,92,95),(63,96,93,66)], [(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)], [(1,94,43,75),(2,93,44,74),(3,92,45,73),(4,91,46,78),(5,96,47,77),(6,95,48,76),(7,79,28,55),(8,84,29,60),(9,83,30,59),(10,82,25,58),(11,81,26,57),(12,80,27,56),(13,85,34,49),(14,90,35,54),(15,89,36,53),(16,88,31,52),(17,87,32,51),(18,86,33,50),(19,42,72,64),(20,41,67,63),(21,40,68,62),(22,39,69,61),(23,38,70,66),(24,37,71,65)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 5 | 0 | 0 | 0 | 0 |
| 10 | 12 | 0 | 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 | 12 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 11 | 3 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 11 |
| 0 | 0 | 0 | 0 | 12 | 5 |
G:=sub<GL(6,GF(13))| [1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,1,5,0,0,0,0,0,0,1,3,0,0,0,0,8,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,2,0,0,0,0,1,5,0,0,0,0,0,0,1,0,0,0,0,0,8,12,0,0,0,0,0,0,9,11,0,0,0,0,0,3],[8,2,0,0,0,0,1,5,0,0,0,0,0,0,5,0,0,0,0,0,1,8,0,0,0,0,0,0,8,12,0,0,0,0,11,5] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 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 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
42 irreducible representations
| dim | 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 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D4 | D4⋊2S3 | D4⋊2S3 | S3×C4○D4 |
| kernel | C4⋊C4.178D6 | C23.16D6 | C23.8D6 | Dic6⋊C4 | C12.3Q8 | C2×C4×Dic3 | C12.48D4 | D4×Dic3 | C23.23D6 | C23.12D6 | C3×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | C2×C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 4 | 4 | 4 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._{178}D_6 % in TeX
G:=Group("C4:C4.178D6"); // GroupNames label
G:=SmallGroup(192,1159);
// by ID
G=gap.SmallGroup(192,1159);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,794,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations