metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.2Dic7, (C4×C28).2C4, (C2×C28).5D4, (D4×C14).2C4, (C2×Q8).3D14, (C2×D4).2Dic7, C4.4D4.3D7, C28.10D4⋊2C2, C7⋊2(C42.C4), (Q8×C14).3C22, C14.24(C23⋊C4), C2.9(C23⋊Dic7), C22.15(C23.D7), (C2×C28).9(C2×C4), (C2×C4).7(C7⋊D4), (C2×C4).2(C2×Dic7), (C7×C4.4D4).1C2, (C2×C14).100(C22⋊C4), SmallGroup(448,99)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.Dic7
G = < a,b,c,d | a4=b4=1, c14=b2, d2=b2c7, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c13 >
Subgroups: 236 in 64 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C28, C2×C14, C2×C14, C4.10D4, C4.4D4, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C42.C4, C4.Dic7, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C28.10D4, C7×C4.4D4, C42.Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C42.C4, C23.D7, C23⋊Dic7, C42.Dic7
►On 112 points
Generators in S112
(1 15)(3 17)(5 19)(7 21)(9 23)(11 25)(13 27)(29 68 43 82)(30 69 44 83)(31 70 45 84)(32 71 46 57)(33 72 47 58)(34 73 48 59)(35 74 49 60)(36 75 50 61)(37 76 51 62)(38 77 52 63)(39 78 53 64)(40 79 54 65)(41 80 55 66)(42 81 56 67)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(1 87 15 101)(2 102 16 88)(3 89 17 103)(4 104 18 90)(5 91 19 105)(6 106 20 92)(7 93 21 107)(8 108 22 94)(9 95 23 109)(10 110 24 96)(11 97 25 111)(12 112 26 98)(13 99 27 85)(14 86 28 100)(29 82 43 68)(30 69 44 83)(31 84 45 70)(32 71 46 57)(33 58 47 72)(34 73 48 59)(35 60 49 74)(36 75 50 61)(37 62 51 76)(38 77 52 63)(39 64 53 78)(40 79 54 65)(41 66 55 80)(42 81 56 67)
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 66 22 59 15 80 8 73)(2 79 23 72 16 65 9 58)(3 64 24 57 17 78 10 71)(4 77 25 70 18 63 11 84)(5 62 26 83 19 76 12 69)(6 75 27 68 20 61 13 82)(7 60 28 81 21 74 14 67)(29 92 50 85 43 106 36 99)(30 105 51 98 44 91 37 112)(31 90 52 111 45 104 38 97)(32 103 53 96 46 89 39 110)(33 88 54 109 47 102 40 95)(34 101 55 94 48 87 41 108)(35 86 56 107 49 100 42 93)
G:=sub<Sym(112)| (1,15)(3,17)(5,19)(7,21)(9,23)(11,25)(13,27)(29,68,43,82)(30,69,44,83)(31,70,45,84)(32,71,46,57)(33,72,47,58)(34,73,48,59)(35,74,49,60)(36,75,50,61)(37,76,51,62)(38,77,52,63)(39,78,53,64)(40,79,54,65)(41,80,55,66)(42,81,56,67)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (1,87,15,101)(2,102,16,88)(3,89,17,103)(4,104,18,90)(5,91,19,105)(6,106,20,92)(7,93,21,107)(8,108,22,94)(9,95,23,109)(10,110,24,96)(11,97,25,111)(12,112,26,98)(13,99,27,85)(14,86,28,100)(29,82,43,68)(30,69,44,83)(31,84,45,70)(32,71,46,57)(33,58,47,72)(34,73,48,59)(35,60,49,74)(36,75,50,61)(37,62,51,76)(38,77,52,63)(39,64,53,78)(40,79,54,65)(41,66,55,80)(42,81,56,67), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,66,22,59,15,80,8,73)(2,79,23,72,16,65,9,58)(3,64,24,57,17,78,10,71)(4,77,25,70,18,63,11,84)(5,62,26,83,19,76,12,69)(6,75,27,68,20,61,13,82)(7,60,28,81,21,74,14,67)(29,92,50,85,43,106,36,99)(30,105,51,98,44,91,37,112)(31,90,52,111,45,104,38,97)(32,103,53,96,46,89,39,110)(33,88,54,109,47,102,40,95)(34,101,55,94,48,87,41,108)(35,86,56,107,49,100,42,93)>;
G:=Group( (1,15)(3,17)(5,19)(7,21)(9,23)(11,25)(13,27)(29,68,43,82)(30,69,44,83)(31,70,45,84)(32,71,46,57)(33,72,47,58)(34,73,48,59)(35,74,49,60)(36,75,50,61)(37,76,51,62)(38,77,52,63)(39,78,53,64)(40,79,54,65)(41,80,55,66)(42,81,56,67)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (1,87,15,101)(2,102,16,88)(3,89,17,103)(4,104,18,90)(5,91,19,105)(6,106,20,92)(7,93,21,107)(8,108,22,94)(9,95,23,109)(10,110,24,96)(11,97,25,111)(12,112,26,98)(13,99,27,85)(14,86,28,100)(29,82,43,68)(30,69,44,83)(31,84,45,70)(32,71,46,57)(33,58,47,72)(34,73,48,59)(35,60,49,74)(36,75,50,61)(37,62,51,76)(38,77,52,63)(39,64,53,78)(40,79,54,65)(41,66,55,80)(42,81,56,67), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,66,22,59,15,80,8,73)(2,79,23,72,16,65,9,58)(3,64,24,57,17,78,10,71)(4,77,25,70,18,63,11,84)(5,62,26,83,19,76,12,69)(6,75,27,68,20,61,13,82)(7,60,28,81,21,74,14,67)(29,92,50,85,43,106,36,99)(30,105,51,98,44,91,37,112)(31,90,52,111,45,104,38,97)(32,103,53,96,46,89,39,110)(33,88,54,109,47,102,40,95)(34,101,55,94,48,87,41,108)(35,86,56,107,49,100,42,93) );
G=PermutationGroup([[(1,15),(3,17),(5,19),(7,21),(9,23),(11,25),(13,27),(29,68,43,82),(30,69,44,83),(31,70,45,84),(32,71,46,57),(33,72,47,58),(34,73,48,59),(35,74,49,60),(36,75,50,61),(37,76,51,62),(38,77,52,63),(39,78,53,64),(40,79,54,65),(41,80,55,66),(42,81,56,67),(85,99),(87,101),(89,103),(91,105),(93,107),(95,109),(97,111)], [(1,87,15,101),(2,102,16,88),(3,89,17,103),(4,104,18,90),(5,91,19,105),(6,106,20,92),(7,93,21,107),(8,108,22,94),(9,95,23,109),(10,110,24,96),(11,97,25,111),(12,112,26,98),(13,99,27,85),(14,86,28,100),(29,82,43,68),(30,69,44,83),(31,84,45,70),(32,71,46,57),(33,58,47,72),(34,73,48,59),(35,60,49,74),(36,75,50,61),(37,62,51,76),(38,77,52,63),(39,64,53,78),(40,79,54,65),(41,66,55,80),(42,81,56,67)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,66,22,59,15,80,8,73),(2,79,23,72,16,65,9,58),(3,64,24,57,17,78,10,71),(4,77,25,70,18,63,11,84),(5,62,26,83,19,76,12,69),(6,75,27,68,20,61,13,82),(7,60,28,81,21,74,14,67),(29,92,50,85,43,106,36,99),(30,105,51,98,44,91,37,112),(31,90,52,111,45,104,38,97),(32,103,53,96,46,89,39,110),(33,88,54,109,47,102,40,95),(34,101,55,94,48,87,41,108),(35,86,56,107,49,100,42,93)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 8 | 4 | ··· | 4 | 2 | 2 | 2 | 56 | 56 | 56 | 56 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D7 | Dic7 | Dic7 | D14 | C7⋊D4 | C23⋊C4 | C42.C4 | C23⋊Dic7 | C42.Dic7 |
| kernel | C42.Dic7 | C28.10D4 | C7×C4.4D4 | C4×C28 | D4×C14 | C2×C28 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C2×C4 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of C42.Dic7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 94 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 98 | 0 | 0 | 0 |
| 54 | 15 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 54 | 15 |
| 16 | 34 | 0 | 0 |
| 78 | 97 | 0 | 0 |
| 0 | 0 | 106 | 84 |
| 0 | 0 | 93 | 7 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 101 | 0 | 0 |
| 19 | 112 | 0 | 0 |
G:=sub<GL(4,GF(113))| [112,94,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[98,54,0,0,0,15,0,0,0,0,98,54,0,0,0,15],[16,78,0,0,34,97,0,0,0,0,106,93,0,0,84,7],[0,0,1,19,0,0,101,112,1,0,0,0,0,1,0,0] >;
C42.Dic7 in GAP, Magma, Sage, TeX
C_4^2.{\rm Dic}_7 % in TeX
G:=Group("C4^2.Dic7"); // GroupNames label
G:=SmallGroup(448,99);
// by ID
G=gap.SmallGroup(448,99);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,184,1571,570,297,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^13>;
// generators/relations