metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.106D6, C6.572- (1+4), C4⋊C4.314D6, (C4×D4).13S3, C12⋊2Q8⋊22C2, (C4×Dic6)⋊28C2, (D4×C12).14C2, (C2×D4).210D6, (C2×C6).86C24, C4.15(C4○D12), C2.15(Q8○D12), C22⋊C4.107D6, (C22×C4).221D6, C23.8D6⋊7C2, Dic6⋊C4⋊14C2, C12.109(C4○D4), C12.48D4⋊19C2, (C2×C12).155C23, (C4×C12).148C22, C23.12D6.7C2, (C6×D4).250C22, C23.26D6⋊7C2, C4.116(D4⋊2S3), C4⋊Dic3.297C22, C22.114(S3×C23), C23.102(C22×S3), (C22×C6).156C23, (C4×Dic3).73C22, (C2×Dic3).36C23, Dic3⋊C4.109C22, (C22×C12).105C22, C3⋊2(C22.50C24), (C2×Dic6).237C22, C6.D4.103C22, C6.38(C2×C4○D4), C2.42(C2×C4○D12), C2.20(C2×D4⋊2S3), (C3×C4⋊C4).322C22, (C2×C4).281(C22×S3), (C3×C22⋊C4).120C22, SmallGroup(192,1101)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 440 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×11], C22, C22 [×6], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], Q8 [×6], C23 [×2], Dic3 [×8], C12 [×4], C12 [×3], C2×C6, C2×C6 [×6], C42, C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×11], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic6 [×6], C2×Dic3 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×2], C22×C6 [×2], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3 [×6], Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×4], C6.D4 [×8], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C2×Dic6 [×2], C22×C12 [×2], C6×D4, C22.50C24, C4×Dic6, C12⋊2Q8, C23.8D6 [×4], Dic6⋊C4 [×2], C12.48D4 [×2], C23.26D6 [×2], C23.12D6 [×2], D4×C12, C42.106D6
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], D4⋊2S3 [×2], S3×C23, C22.50C24, C2×C4○D12, C2×D4⋊2S3, Q8○D12, C42.106D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 36 74 40)(2 41 75 25)(3 26 76 42)(4 43 77 27)(5 28 78 44)(6 45 79 29)(7 30 80 46)(8 47 81 31)(9 32 82 48)(10 37 83 33)(11 34 84 38)(12 39 73 35)(13 86 63 59)(14 60 64 87)(15 88 65 49)(16 50 66 89)(17 90 67 51)(18 52 68 91)(19 92 69 53)(20 54 70 93)(21 94 71 55)(22 56 72 95)(23 96 61 57)(24 58 62 85)
(1 27 7 33)(2 28 8 34)(3 29 9 35)(4 30 10 36)(5 31 11 25)(6 32 12 26)(13 95 19 89)(14 96 20 90)(15 85 21 91)(16 86 22 92)(17 87 23 93)(18 88 24 94)(37 74 43 80)(38 75 44 81)(39 76 45 82)(40 77 46 83)(41 78 47 84)(42 79 48 73)(49 62 55 68)(50 63 56 69)(51 64 57 70)(52 65 58 71)(53 66 59 72)(54 67 60 61)
(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 88 74 49)(2 93 75 54)(3 86 76 59)(4 91 77 52)(5 96 78 57)(6 89 79 50)(7 94 80 55)(8 87 81 60)(9 92 82 53)(10 85 83 58)(11 90 84 51)(12 95 73 56)(13 42 63 26)(14 47 64 31)(15 40 65 36)(16 45 66 29)(17 38 67 34)(18 43 68 27)(19 48 69 32)(20 41 70 25)(21 46 71 30)(22 39 72 35)(23 44 61 28)(24 37 62 33)
G:=sub<Sym(96)| (1,36,74,40)(2,41,75,25)(3,26,76,42)(4,43,77,27)(5,28,78,44)(6,45,79,29)(7,30,80,46)(8,47,81,31)(9,32,82,48)(10,37,83,33)(11,34,84,38)(12,39,73,35)(13,86,63,59)(14,60,64,87)(15,88,65,49)(16,50,66,89)(17,90,67,51)(18,52,68,91)(19,92,69,53)(20,54,70,93)(21,94,71,55)(22,56,72,95)(23,96,61,57)(24,58,62,85), (1,27,7,33)(2,28,8,34)(3,29,9,35)(4,30,10,36)(5,31,11,25)(6,32,12,26)(13,95,19,89)(14,96,20,90)(15,85,21,91)(16,86,22,92)(17,87,23,93)(18,88,24,94)(37,74,43,80)(38,75,44,81)(39,76,45,82)(40,77,46,83)(41,78,47,84)(42,79,48,73)(49,62,55,68)(50,63,56,69)(51,64,57,70)(52,65,58,71)(53,66,59,72)(54,67,60,61), (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), (1,88,74,49)(2,93,75,54)(3,86,76,59)(4,91,77,52)(5,96,78,57)(6,89,79,50)(7,94,80,55)(8,87,81,60)(9,92,82,53)(10,85,83,58)(11,90,84,51)(12,95,73,56)(13,42,63,26)(14,47,64,31)(15,40,65,36)(16,45,66,29)(17,38,67,34)(18,43,68,27)(19,48,69,32)(20,41,70,25)(21,46,71,30)(22,39,72,35)(23,44,61,28)(24,37,62,33)>;
G:=Group( (1,36,74,40)(2,41,75,25)(3,26,76,42)(4,43,77,27)(5,28,78,44)(6,45,79,29)(7,30,80,46)(8,47,81,31)(9,32,82,48)(10,37,83,33)(11,34,84,38)(12,39,73,35)(13,86,63,59)(14,60,64,87)(15,88,65,49)(16,50,66,89)(17,90,67,51)(18,52,68,91)(19,92,69,53)(20,54,70,93)(21,94,71,55)(22,56,72,95)(23,96,61,57)(24,58,62,85), (1,27,7,33)(2,28,8,34)(3,29,9,35)(4,30,10,36)(5,31,11,25)(6,32,12,26)(13,95,19,89)(14,96,20,90)(15,85,21,91)(16,86,22,92)(17,87,23,93)(18,88,24,94)(37,74,43,80)(38,75,44,81)(39,76,45,82)(40,77,46,83)(41,78,47,84)(42,79,48,73)(49,62,55,68)(50,63,56,69)(51,64,57,70)(52,65,58,71)(53,66,59,72)(54,67,60,61), (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), (1,88,74,49)(2,93,75,54)(3,86,76,59)(4,91,77,52)(5,96,78,57)(6,89,79,50)(7,94,80,55)(8,87,81,60)(9,92,82,53)(10,85,83,58)(11,90,84,51)(12,95,73,56)(13,42,63,26)(14,47,64,31)(15,40,65,36)(16,45,66,29)(17,38,67,34)(18,43,68,27)(19,48,69,32)(20,41,70,25)(21,46,71,30)(22,39,72,35)(23,44,61,28)(24,37,62,33) );
G=PermutationGroup([(1,36,74,40),(2,41,75,25),(3,26,76,42),(4,43,77,27),(5,28,78,44),(6,45,79,29),(7,30,80,46),(8,47,81,31),(9,32,82,48),(10,37,83,33),(11,34,84,38),(12,39,73,35),(13,86,63,59),(14,60,64,87),(15,88,65,49),(16,50,66,89),(17,90,67,51),(18,52,68,91),(19,92,69,53),(20,54,70,93),(21,94,71,55),(22,56,72,95),(23,96,61,57),(24,58,62,85)], [(1,27,7,33),(2,28,8,34),(3,29,9,35),(4,30,10,36),(5,31,11,25),(6,32,12,26),(13,95,19,89),(14,96,20,90),(15,85,21,91),(16,86,22,92),(17,87,23,93),(18,88,24,94),(37,74,43,80),(38,75,44,81),(39,76,45,82),(40,77,46,83),(41,78,47,84),(42,79,48,73),(49,62,55,68),(50,63,56,69),(51,64,57,70),(52,65,58,71),(53,66,59,72),(54,67,60,61)], [(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)], [(1,88,74,49),(2,93,75,54),(3,86,76,59),(4,91,77,52),(5,96,78,57),(6,89,79,50),(7,94,80,55),(8,87,81,60),(9,92,82,53),(10,85,83,58),(11,90,84,51),(12,95,73,56),(13,42,63,26),(14,47,64,31),(15,40,65,36),(16,45,66,29),(17,38,67,34),(18,43,68,27),(19,48,69,32),(20,41,70,25),(21,46,71,30),(22,39,72,35),(23,44,61,28),(24,37,62,33)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 5 | 12 |
| 5 | 4 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 1 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
| 10 | 5 | 0 | 0 |
| 1 | 3 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,5,0,0,10,12],[5,0,0,0,4,8,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,1,11,0,0,0,0,12,0,0,0,3,1],[10,1,0,0,5,3,0,0,0,0,8,0,0,0,2,5] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4S | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 2 | ··· | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- (1+4) | D4⋊2S3 | Q8○D12 |
| kernel | C42.106D6 | C4×Dic6 | C12⋊2Q8 | C23.8D6 | Dic6⋊C4 | C12.48D4 | C23.26D6 | C23.12D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{106}D_6 % in TeX
G:=Group("C4^2.106D6"); // GroupNames label
G:=SmallGroup(192,1101);
// by ID
G=gap.SmallGroup(192,1101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,1571,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations