direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7xD8:2C4, D8:2C28, Q16:2C28, C56.102D4, M5(2):5C14, C28.43SD16, (C7xD8):8C4, C8.1(C2xC28), (C7xQ16):8C4, C4.Q8:1C14, C8.22(C7xD4), C56.42(C2xC4), C4oD8.2C14, (C2xC14).25D8, C4.8(C7xSD16), C22.3(C7xD8), (C2xC28).279D4, (C7xM5(2)):13C2, C28.72(C22:C4), (C2xC56).265C22, C14.40(D4:C4), (C7xC4oD8).7C2, (C7xC4.Q8):10C2, (C2xC4).10(C7xD4), C4.4(C7xC22:C4), (C2xC8).12(C2xC14), C2.9(C7xD4:C4), SmallGroup(448,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7xD8:2C4
G = < a,b,c,d | a7=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >
Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, Q8, C14, C14, C16, C4:C4, C2xC8, D8, SD16, Q16, C4oD4, C28, C28, C2xC14, C2xC14, C4.Q8, M5(2), C4oD8, C56, C2xC28, C2xC28, C7xD4, C7xQ8, D8:2C4, C112, C7xC4:C4, C2xC56, C7xD8, C7xSD16, C7xQ16, C7xC4oD4, C7xC4.Q8, C7xM5(2), C7xC4oD8, C7xD8:2C4
Quotients: C1, C2, C4, C22, C7, C2xC4, D4, C14, C22:C4, D8, SD16, C28, C2xC14, D4:C4, C2xC28, C7xD4, D8:2C4, C7xC22:C4, C7xD8, C7xSD16, C7xD4:C4, C7xD8:2C4
►On 112 points
Generators in S112
(1 55 47 39 31 23 15)(2 56 48 40 32 24 16)(3 49 41 33 25 17 9)(4 50 42 34 26 18 10)(5 51 43 35 27 19 11)(6 52 44 36 28 20 12)(7 53 45 37 29 21 13)(8 54 46 38 30 22 14)(57 105 97 89 81 73 65)(58 106 98 90 82 74 66)(59 107 99 91 83 75 67)(60 108 100 92 84 76 68)(61 109 101 93 85 77 69)(62 110 102 94 86 78 70)(63 111 103 95 87 79 71)(64 112 104 96 88 80 72)
(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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 72)(16 71)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 80)(24 79)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 88)(32 87)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 96)(40 95)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 104)(48 103)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 112)(56 111)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)(57 58 61 62)(59 64 63 60)(65 66 69 70)(67 72 71 68)(73 74 77 78)(75 80 79 76)(81 82 85 86)(83 88 87 84)(89 90 93 94)(91 96 95 92)(97 98 101 102)(99 104 103 100)(105 106 109 110)(107 112 111 108)
G:=sub<Sym(112)| (1,55,47,39,31,23,15)(2,56,48,40,32,24,16)(3,49,41,33,25,17,9)(4,50,42,34,26,18,10)(5,51,43,35,27,19,11)(6,52,44,36,28,20,12)(7,53,45,37,29,21,13)(8,54,46,38,30,22,14)(57,105,97,89,81,73,65)(58,106,98,90,82,74,66)(59,107,99,91,83,75,67)(60,108,100,92,84,76,68)(61,109,101,93,85,77,69)(62,110,102,94,86,78,70)(63,111,103,95,87,79,71)(64,112,104,96,88,80,72), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,72)(16,71)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,80)(24,79)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,88)(32,87)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,96)(40,95)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,104)(48,103)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,112)(56,111), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54)(57,58,61,62)(59,64,63,60)(65,66,69,70)(67,72,71,68)(73,74,77,78)(75,80,79,76)(81,82,85,86)(83,88,87,84)(89,90,93,94)(91,96,95,92)(97,98,101,102)(99,104,103,100)(105,106,109,110)(107,112,111,108)>;
G:=Group( (1,55,47,39,31,23,15)(2,56,48,40,32,24,16)(3,49,41,33,25,17,9)(4,50,42,34,26,18,10)(5,51,43,35,27,19,11)(6,52,44,36,28,20,12)(7,53,45,37,29,21,13)(8,54,46,38,30,22,14)(57,105,97,89,81,73,65)(58,106,98,90,82,74,66)(59,107,99,91,83,75,67)(60,108,100,92,84,76,68)(61,109,101,93,85,77,69)(62,110,102,94,86,78,70)(63,111,103,95,87,79,71)(64,112,104,96,88,80,72), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,72)(16,71)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,80)(24,79)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,88)(32,87)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,96)(40,95)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,104)(48,103)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,112)(56,111), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54)(57,58,61,62)(59,64,63,60)(65,66,69,70)(67,72,71,68)(73,74,77,78)(75,80,79,76)(81,82,85,86)(83,88,87,84)(89,90,93,94)(91,96,95,92)(97,98,101,102)(99,104,103,100)(105,106,109,110)(107,112,111,108) );
G=PermutationGroup([[(1,55,47,39,31,23,15),(2,56,48,40,32,24,16),(3,49,41,33,25,17,9),(4,50,42,34,26,18,10),(5,51,43,35,27,19,11),(6,52,44,36,28,20,12),(7,53,45,37,29,21,13),(8,54,46,38,30,22,14),(57,105,97,89,81,73,65),(58,106,98,90,82,74,66),(59,107,99,91,83,75,67),(60,108,100,92,84,76,68),(61,109,101,93,85,77,69),(62,110,102,94,86,78,70),(63,111,103,95,87,79,71),(64,112,104,96,88,80,72)], [(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)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,72),(16,71),(17,78),(18,77),(19,76),(20,75),(21,74),(22,73),(23,80),(24,79),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,88),(32,87),(33,94),(34,93),(35,92),(36,91),(37,90),(38,89),(39,96),(40,95),(41,102),(42,101),(43,100),(44,99),(45,98),(46,97),(47,104),(48,103),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,112),(56,111)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,21),(18,24),(20,22),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(41,45),(42,48),(44,46),(49,53),(50,56),(52,54),(57,58,61,62),(59,64,63,60),(65,66,69,70),(67,72,71,68),(73,74,77,78),(75,80,79,76),(81,82,85,86),(83,88,87,84),(89,90,93,94),(91,96,95,92),(97,98,101,102),(99,104,103,100),(105,106,109,110),(107,112,111,108)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 16A | 16B | 16C | 16D | 28A | ··· | 28L | 28M | ··· | 28AD | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | D4 | D4 | SD16 | D8 | C7xD4 | C7xD4 | C7xSD16 | C7xD8 | D8:2C4 | C7xD8:2C4 |
| kernel | C7xD8:2C4 | C7xC4.Q8 | C7xM5(2) | C7xC4oD8 | C7xD8 | C7xQ16 | D8:2C4 | C4.Q8 | M5(2) | C4oD8 | D8 | Q16 | C56 | C2xC28 | C28 | C2xC14 | C8 | C2xC4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C7xD8:2C4 ►in GL6(F113)
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 13 | 0 | 0 |
| 0 | 0 | 100 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 100 | 13 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 100 | 13 |
| 0 | 0 | 100 | 13 | 0 | 0 |
| 0 | 0 | 100 | 100 | 0 | 0 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 100 | 100 |
G:=sub<GL(6,GF(113))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,100,100,0,0,0,0,13,100,0,0,0,0,0,0,13,100,0,0,0,0,13,13],[0,112,0,0,0,0,112,0,0,0,0,0,0,0,0,0,100,100,0,0,0,0,13,100,0,0,13,100,0,0,0,0,13,13,0,0],[98,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,13,100,0,0,0,0,100,100] >;
C7xD8:2C4 in GAP, Magma, Sage, TeX
C_7\times D_8\rtimes_2C_4
% in TeX
G:=Group("C7xD8:2C4"); // GroupNames label
G:=SmallGroup(448,164);
// by ID
G=gap.SmallGroup(448,164);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,5106,136,4911,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations