metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.114D6, C6.202+ (1+4), (C4×D4)⋊21S3, (D4×C12)⋊23C2, C4⋊C4.319D6, (C4×Dic6)⋊34C2, (C2×D4).220D6, C23.9D6⋊8C2, C12⋊3D4.7C2, Dic3⋊5D4⋊16C2, (C22×C4).64D6, C42⋊7S3⋊19C2, C4.16(C4○D12), C23.12D6⋊9C2, (C2×C6).103C24, D6⋊C4.87C22, C22⋊C4.116D6, Dic6⋊C4⋊16C2, C12.111(C4○D4), C2.21(D4⋊6D6), (C4×C12).158C22, (C2×C12).701C23, C23.11D6⋊8C2, (C6×D4).263C22, Dic3.35(C4○D4), (C2×D12).139C22, C23.28D6⋊18C2, Dic3⋊C4.66C22, (C22×S3).37C23, C4⋊Dic3.301C22, C22.128(S3×C23), (C22×C6).173C23, C23.110(C22×S3), (C2×Dic3).44C23, (C4×Dic3).76C22, (C22×C12).365C22, C3⋊1(C22.53C24), (C2×Dic6).145C22, C6.D4.107C22, (C4×C3⋊D4)⋊45C2, C6.45(C2×C4○D4), C2.26(S3×C4○D4), C2.52(C2×C4○D12), (S3×C2×C4).202C22, (C3×C4⋊C4).332C22, (C2×C4).286(C22×S3), (C2×C3⋊D4).116C22, (C3×C22⋊C4).127C22, SmallGroup(192,1118)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 600 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×11], C22, C22 [×12], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×4], D6 [×6], C2×C6, C2×C6 [×6], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic6 [×4], C4×S3 [×2], D12 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4×D4, C4×D4 [×3], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C4⋊1D4, C4×Dic3 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×6], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, C22.53C24, C4×Dic6, C42⋊7S3, C23.9D6 [×2], C23.11D6 [×2], Dic6⋊C4, Dic3⋊5D4, C4×C3⋊D4 [×2], C23.28D6 [×2], C23.12D6, C12⋊3D4, D4×C12, C42.114D6
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, C22.53C24, C2×C4○D12, D4⋊6D6, S3×C4○D4, C42.114D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 88 73 30)(2 25 74 95)(3 90 75 32)(4 27 76 85)(5 92 77 34)(6 29 78 87)(7 94 79 36)(8 31 80 89)(9 96 81 26)(10 33 82 91)(11 86 83 28)(12 35 84 93)(13 42 58 72)(14 67 59 37)(15 44 60 62)(16 69 49 39)(17 46 50 64)(18 71 51 41)(19 48 52 66)(20 61 53 43)(21 38 54 68)(22 63 55 45)(23 40 56 70)(24 65 57 47)
(1 71 7 65)(2 72 8 66)(3 61 9 67)(4 62 10 68)(5 63 11 69)(6 64 12 70)(13 31 19 25)(14 32 20 26)(15 33 21 27)(16 34 22 28)(17 35 23 29)(18 36 24 30)(37 75 43 81)(38 76 44 82)(39 77 45 83)(40 78 46 84)(41 79 47 73)(42 80 48 74)(49 92 55 86)(50 93 56 87)(51 94 57 88)(52 95 58 89)(53 96 59 90)(54 85 60 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 49 19 55)(14 54 20 60)(15 59 21 53)(16 52 22 58)(17 57 23 51)(18 50 24 56)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 68 43 62)(38 61 44 67)(39 66 45 72)(40 71 46 65)(41 64 47 70)(42 69 48 63)(73 78 79 84)(74 83 80 77)(75 76 81 82)(85 96 91 90)(86 89 92 95)(87 94 93 88)
G:=sub<Sym(96)| (1,88,73,30)(2,25,74,95)(3,90,75,32)(4,27,76,85)(5,92,77,34)(6,29,78,87)(7,94,79,36)(8,31,80,89)(9,96,81,26)(10,33,82,91)(11,86,83,28)(12,35,84,93)(13,42,58,72)(14,67,59,37)(15,44,60,62)(16,69,49,39)(17,46,50,64)(18,71,51,41)(19,48,52,66)(20,61,53,43)(21,38,54,68)(22,63,55,45)(23,40,56,70)(24,65,57,47), (1,71,7,65)(2,72,8,66)(3,61,9,67)(4,62,10,68)(5,63,11,69)(6,64,12,70)(13,31,19,25)(14,32,20,26)(15,33,21,27)(16,34,22,28)(17,35,23,29)(18,36,24,30)(37,75,43,81)(38,76,44,82)(39,77,45,83)(40,78,46,84)(41,79,47,73)(42,80,48,74)(49,92,55,86)(50,93,56,87)(51,94,57,88)(52,95,58,89)(53,96,59,90)(54,85,60,91), (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)(13,49,19,55)(14,54,20,60)(15,59,21,53)(16,52,22,58)(17,57,23,51)(18,50,24,56)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,68,43,62)(38,61,44,67)(39,66,45,72)(40,71,46,65)(41,64,47,70)(42,69,48,63)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,96,91,90)(86,89,92,95)(87,94,93,88)>;
G:=Group( (1,88,73,30)(2,25,74,95)(3,90,75,32)(4,27,76,85)(5,92,77,34)(6,29,78,87)(7,94,79,36)(8,31,80,89)(9,96,81,26)(10,33,82,91)(11,86,83,28)(12,35,84,93)(13,42,58,72)(14,67,59,37)(15,44,60,62)(16,69,49,39)(17,46,50,64)(18,71,51,41)(19,48,52,66)(20,61,53,43)(21,38,54,68)(22,63,55,45)(23,40,56,70)(24,65,57,47), (1,71,7,65)(2,72,8,66)(3,61,9,67)(4,62,10,68)(5,63,11,69)(6,64,12,70)(13,31,19,25)(14,32,20,26)(15,33,21,27)(16,34,22,28)(17,35,23,29)(18,36,24,30)(37,75,43,81)(38,76,44,82)(39,77,45,83)(40,78,46,84)(41,79,47,73)(42,80,48,74)(49,92,55,86)(50,93,56,87)(51,94,57,88)(52,95,58,89)(53,96,59,90)(54,85,60,91), (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)(13,49,19,55)(14,54,20,60)(15,59,21,53)(16,52,22,58)(17,57,23,51)(18,50,24,56)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,68,43,62)(38,61,44,67)(39,66,45,72)(40,71,46,65)(41,64,47,70)(42,69,48,63)(73,78,79,84)(74,83,80,77)(75,76,81,82)(85,96,91,90)(86,89,92,95)(87,94,93,88) );
G=PermutationGroup([(1,88,73,30),(2,25,74,95),(3,90,75,32),(4,27,76,85),(5,92,77,34),(6,29,78,87),(7,94,79,36),(8,31,80,89),(9,96,81,26),(10,33,82,91),(11,86,83,28),(12,35,84,93),(13,42,58,72),(14,67,59,37),(15,44,60,62),(16,69,49,39),(17,46,50,64),(18,71,51,41),(19,48,52,66),(20,61,53,43),(21,38,54,68),(22,63,55,45),(23,40,56,70),(24,65,57,47)], [(1,71,7,65),(2,72,8,66),(3,61,9,67),(4,62,10,68),(5,63,11,69),(6,64,12,70),(13,31,19,25),(14,32,20,26),(15,33,21,27),(16,34,22,28),(17,35,23,29),(18,36,24,30),(37,75,43,81),(38,76,44,82),(39,77,45,83),(40,78,46,84),(41,79,47,73),(42,80,48,74),(49,92,55,86),(50,93,56,87),(51,94,57,88),(52,95,58,89),(53,96,59,90),(54,85,60,91)], [(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),(13,49,19,55),(14,54,20,60),(15,59,21,53),(16,52,22,58),(17,57,23,51),(18,50,24,56),(25,28,31,34),(26,33,32,27),(29,36,35,30),(37,68,43,62),(38,61,44,67),(39,66,45,72),(40,71,46,65),(41,64,47,70),(42,69,48,63),(73,78,79,84),(74,83,80,77),(75,76,81,82),(85,96,91,90),(86,89,92,95),(87,94,93,88)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 5 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 8 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [0,5,0,0,8,0,0,0,0,0,3,7,0,0,6,10],[8,0,0,0,0,8,0,0,0,0,3,7,0,0,6,10],[5,0,0,0,0,8,0,0,0,0,5,8,0,0,5,0],[5,0,0,0,0,5,0,0,0,0,5,0,0,0,5,8] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 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 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 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 | 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 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ (1+4) | D4⋊6D6 | S3×C4○D4 |
| kernel | C42.114D6 | C4×Dic6 | C42⋊7S3 | C23.9D6 | C23.11D6 | Dic6⋊C4 | Dic3⋊5D4 | C4×C3⋊D4 | C23.28D6 | C23.12D6 | C12⋊3D4 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{114}D_6 % in TeX
G:=Group("C4^2.114D6"); // GroupNames label
G:=SmallGroup(192,1118);
// by ID
G=gap.SmallGroup(192,1118);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,136,6278]);
// 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^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations