metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊2Dic7, (C4×C28)⋊2C4, (Q8×C14)⋊2C4, (C2×Q8)⋊2Dic7, (C2×D4).9D14, C7⋊2(C42⋊3C4), C4.4D4.2D7, (C22×C14).16D4, C23.7(C7⋊D4), C23⋊Dic7.4C2, C14.23(C23⋊C4), C2.8(C23⋊Dic7), (D4×C14).172C22, C22.14(C23.D7), (C2×C28).8(C2×C4), (C2×C4).1(C2×Dic7), (C7×C4.4D4).9C2, (C2×C14).99(C22⋊C4), SmallGroup(448,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊2Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 364 in 70 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C4.4D4, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C42⋊3C4, C23.D7, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C23⋊Dic7, C7×C4.4D4, C42⋊2Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C42⋊3C4, C23.D7, C23⋊Dic7, C42⋊2Dic7
►On 112 points
Generators in S112
(1 98)(3 86)(5 88)(7 90)(9 92)(11 94)(13 96)(16 72)(18 74)(20 76)(22 78)(24 80)(26 82)(28 84)(29 53 110 57)(30 54 111 58)(31 55 112 59)(32 56 99 60)(33 43 100 61)(34 44 101 62)(35 45 102 63)(36 46 103 64)(37 47 104 65)(38 48 105 66)(39 49 106 67)(40 50 107 68)(41 51 108 69)(42 52 109 70)
(1 76 98 20)(2 21 85 77)(3 78 86 22)(4 23 87 79)(5 80 88 24)(6 25 89 81)(7 82 90 26)(8 27 91 83)(9 84 92 28)(10 15 93 71)(11 72 94 16)(12 17 95 73)(13 74 96 18)(14 19 97 75)(29 53 110 57)(30 58 111 54)(31 55 112 59)(32 60 99 56)(33 43 100 61)(34 62 101 44)(35 45 102 63)(36 64 103 46)(37 47 104 65)(38 66 105 48)(39 49 106 67)(40 68 107 50)(41 51 108 69)(42 70 109 52)
(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 48 8 55)(2 47 9 54)(3 46 10 53)(4 45 11 52)(5 44 12 51)(6 43 13 50)(7 56 14 49)(15 29 22 36)(16 42 23 35)(17 41 24 34)(18 40 25 33)(19 39 26 32)(20 38 27 31)(21 37 28 30)(57 86 64 93)(58 85 65 92)(59 98 66 91)(60 97 67 90)(61 96 68 89)(62 95 69 88)(63 94 70 87)(71 110 78 103)(72 109 79 102)(73 108 80 101)(74 107 81 100)(75 106 82 99)(76 105 83 112)(77 104 84 111)
G:=sub<Sym(112)| (1,98)(3,86)(5,88)(7,90)(9,92)(11,94)(13,96)(16,72)(18,74)(20,76)(22,78)(24,80)(26,82)(28,84)(29,53,110,57)(30,54,111,58)(31,55,112,59)(32,56,99,60)(33,43,100,61)(34,44,101,62)(35,45,102,63)(36,46,103,64)(37,47,104,65)(38,48,105,66)(39,49,106,67)(40,50,107,68)(41,51,108,69)(42,52,109,70), (1,76,98,20)(2,21,85,77)(3,78,86,22)(4,23,87,79)(5,80,88,24)(6,25,89,81)(7,82,90,26)(8,27,91,83)(9,84,92,28)(10,15,93,71)(11,72,94,16)(12,17,95,73)(13,74,96,18)(14,19,97,75)(29,53,110,57)(30,58,111,54)(31,55,112,59)(32,60,99,56)(33,43,100,61)(34,62,101,44)(35,45,102,63)(36,64,103,46)(37,47,104,65)(38,66,105,48)(39,49,106,67)(40,68,107,50)(41,51,108,69)(42,70,109,52), (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), (1,48,8,55)(2,47,9,54)(3,46,10,53)(4,45,11,52)(5,44,12,51)(6,43,13,50)(7,56,14,49)(15,29,22,36)(16,42,23,35)(17,41,24,34)(18,40,25,33)(19,39,26,32)(20,38,27,31)(21,37,28,30)(57,86,64,93)(58,85,65,92)(59,98,66,91)(60,97,67,90)(61,96,68,89)(62,95,69,88)(63,94,70,87)(71,110,78,103)(72,109,79,102)(73,108,80,101)(74,107,81,100)(75,106,82,99)(76,105,83,112)(77,104,84,111)>;
G:=Group( (1,98)(3,86)(5,88)(7,90)(9,92)(11,94)(13,96)(16,72)(18,74)(20,76)(22,78)(24,80)(26,82)(28,84)(29,53,110,57)(30,54,111,58)(31,55,112,59)(32,56,99,60)(33,43,100,61)(34,44,101,62)(35,45,102,63)(36,46,103,64)(37,47,104,65)(38,48,105,66)(39,49,106,67)(40,50,107,68)(41,51,108,69)(42,52,109,70), (1,76,98,20)(2,21,85,77)(3,78,86,22)(4,23,87,79)(5,80,88,24)(6,25,89,81)(7,82,90,26)(8,27,91,83)(9,84,92,28)(10,15,93,71)(11,72,94,16)(12,17,95,73)(13,74,96,18)(14,19,97,75)(29,53,110,57)(30,58,111,54)(31,55,112,59)(32,60,99,56)(33,43,100,61)(34,62,101,44)(35,45,102,63)(36,64,103,46)(37,47,104,65)(38,66,105,48)(39,49,106,67)(40,68,107,50)(41,51,108,69)(42,70,109,52), (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), (1,48,8,55)(2,47,9,54)(3,46,10,53)(4,45,11,52)(5,44,12,51)(6,43,13,50)(7,56,14,49)(15,29,22,36)(16,42,23,35)(17,41,24,34)(18,40,25,33)(19,39,26,32)(20,38,27,31)(21,37,28,30)(57,86,64,93)(58,85,65,92)(59,98,66,91)(60,97,67,90)(61,96,68,89)(62,95,69,88)(63,94,70,87)(71,110,78,103)(72,109,79,102)(73,108,80,101)(74,107,81,100)(75,106,82,99)(76,105,83,112)(77,104,84,111) );
G=PermutationGroup([[(1,98),(3,86),(5,88),(7,90),(9,92),(11,94),(13,96),(16,72),(18,74),(20,76),(22,78),(24,80),(26,82),(28,84),(29,53,110,57),(30,54,111,58),(31,55,112,59),(32,56,99,60),(33,43,100,61),(34,44,101,62),(35,45,102,63),(36,46,103,64),(37,47,104,65),(38,48,105,66),(39,49,106,67),(40,50,107,68),(41,51,108,69),(42,52,109,70)], [(1,76,98,20),(2,21,85,77),(3,78,86,22),(4,23,87,79),(5,80,88,24),(6,25,89,81),(7,82,90,26),(8,27,91,83),(9,84,92,28),(10,15,93,71),(11,72,94,16),(12,17,95,73),(13,74,96,18),(14,19,97,75),(29,53,110,57),(30,58,111,54),(31,55,112,59),(32,60,99,56),(33,43,100,61),(34,62,101,44),(35,45,102,63),(36,64,103,46),(37,47,104,65),(38,66,105,48),(39,49,106,67),(40,68,107,50),(41,51,108,69),(42,70,109,52)], [(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)], [(1,48,8,55),(2,47,9,54),(3,46,10,53),(4,45,11,52),(5,44,12,51),(6,43,13,50),(7,56,14,49),(15,29,22,36),(16,42,23,35),(17,41,24,34),(18,40,25,33),(19,39,26,32),(20,38,27,31),(21,37,28,30),(57,86,64,93),(58,85,65,92),(59,98,66,91),(60,97,67,90),(61,96,68,89),(62,95,69,88),(63,94,70,87),(71,110,78,103),(72,109,79,102),(73,108,80,101),(74,107,81,100),(75,106,82,99),(76,105,83,112),(77,104,84,111)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 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 | D14 | Dic7 | C7⋊D4 | C23⋊C4 | C42⋊3C4 | C23⋊Dic7 | C42⋊2Dic7 |
| kernel | C42⋊2Dic7 | C23⋊Dic7 | C7×C4.4D4 | C4×C28 | Q8×C14 | C22×C14 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C23 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of C42⋊2Dic7 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 17 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 20 | 0 | 0 |
| 20 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,17,0,0,0,0,17],[17,0,0,0,0,12,0,0,0,0,17,0,0,0,0,12],[0,20,0,0,20,0,0,0,0,0,0,16,0,0,16,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C42⋊2Dic7 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2{\rm Dic}_7 % in TeX
G:=Group("C4^2:2Dic7"); // GroupNames label
G:=SmallGroup(448,98);
// by ID
G=gap.SmallGroup(448,98);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,219,1571,570,297,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^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^-1>;
// generators/relations