direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D8⋊C22, C28.84C24, C56.51C23, C4○D8⋊3C14, D8⋊4(C2×C14), C8⋊C22⋊6C14, Q16⋊4(C2×C14), C4.68(D4×C14), (C2×C56)⋊30C22, SD16⋊3(C2×C14), (C2×C28).527D4, C28.473(C2×D4), (C7×D8)⋊20C22, C8.C22⋊6C14, C8.2(C22×C14), C4.7(C23×C14), C23.20(C7×D4), (D4×C14)⋊67C22, (C2×M4(2))⋊5C14, M4(2)⋊5(C2×C14), (Q8×C14)⋊56C22, (C7×Q16)⋊18C22, (C7×D4).37C23, D4.4(C22×C14), (C22×C14).38D4, C22.25(D4×C14), Q8.4(C22×C14), (C7×Q8).38C23, (C14×M4(2))⋊15C2, (C2×C28).686C23, (C7×SD16)⋊19C22, C14.205(C22×D4), (C7×M4(2))⋊31C22, (C22×C28).467C22, (C2×C8)⋊3(C2×C14), C2.29(D4×C2×C14), C4○D4⋊5(C2×C14), (C7×C4○D8)⋊10C2, (C14×C4○D4)⋊28C2, (C2×C4○D4)⋊12C14, (C2×D4)⋊16(C2×C14), (C7×C8⋊C22)⋊13C2, (C2×Q8)⋊16(C2×C14), (C2×C4).138(C7×D4), (C2×C14).421(C2×D4), (C7×C4○D4)⋊25C22, (C7×C8.C22)⋊13C2, (C2×C4).47(C22×C14), (C22×C4).78(C2×C14), SmallGroup(448,1358)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 262 conjugacy classes, 158 normal (22 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C14, C14 [×7], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C28 [×2], C28 [×2], C28 [×4], C2×C14, C2×C14 [×2], C2×C14 [×9], C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C56 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×10], C7×D4 [×4], C7×D4 [×10], C7×Q8 [×4], C7×Q8 [×2], C22×C14, C22×C14 [×2], D8⋊C22, C2×C56 [×2], C7×M4(2) [×4], C7×D8 [×4], C7×SD16 [×8], C7×Q16 [×4], C22×C28, C22×C28 [×2], D4×C14 [×2], D4×C14 [×2], Q8×C14 [×2], C7×C4○D4 [×8], C7×C4○D4 [×4], C14×M4(2), C7×C4○D8 [×4], C7×C8⋊C22 [×4], C7×C8.C22 [×4], C14×C4○D4 [×2], C7×D8⋊C22
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C24, C2×C14 [×35], C22×D4, C7×D4 [×4], C22×C14 [×15], D8⋊C22, D4×C14 [×6], C23×C14, D4×C2×C14, C7×D8⋊C22
Generators and relations
G = < a,b,c,d,e | a7=b8=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd=b5, be=eb, dcd=ece=b4c, de=ed >
►On 112 points
(1 75 42 18 91 67 34)(2 76 43 19 92 68 35)(3 77 44 20 93 69 36)(4 78 45 21 94 70 37)(5 79 46 22 95 71 38)(6 80 47 23 96 72 39)(7 73 48 24 89 65 40)(8 74 41 17 90 66 33)(9 106 82 49 25 98 61)(10 107 83 50 26 99 62)(11 108 84 51 27 100 63)(12 109 85 52 28 101 64)(13 110 86 53 29 102 57)(14 111 87 54 30 103 58)(15 112 88 55 31 104 59)(16 105 81 56 32 97 60)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)(33 34)(35 40)(36 39)(37 38)(41 42)(43 48)(44 47)(45 46)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(109 112)(110 111)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(106 110)(108 112)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 112)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(49 68)(50 69)(51 70)(52 71)(53 72)(54 65)(55 66)(56 67)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 89)(88 90)
G:=sub<Sym(112)| (1,75,42,18,91,67,34)(2,76,43,19,92,68,35)(3,77,44,20,93,69,36)(4,78,45,21,94,70,37)(5,79,46,22,95,71,38)(6,80,47,23,96,72,39)(7,73,48,24,89,65,40)(8,74,41,17,90,66,33)(9,106,82,49,25,98,61)(10,107,83,50,26,99,62)(11,108,84,51,27,100,63)(12,109,85,52,28,101,64)(13,110,86,53,29,102,57)(14,111,87,54,30,103,58)(15,112,88,55,31,104,59)(16,105,81,56,32,97,60), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(98,102)(100,104)(106,110)(108,112), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(49,68)(50,69)(51,70)(52,71)(53,72)(54,65)(55,66)(56,67)(57,80)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,89)(88,90)>;
G:=Group( (1,75,42,18,91,67,34)(2,76,43,19,92,68,35)(3,77,44,20,93,69,36)(4,78,45,21,94,70,37)(5,79,46,22,95,71,38)(6,80,47,23,96,72,39)(7,73,48,24,89,65,40)(8,74,41,17,90,66,33)(9,106,82,49,25,98,61)(10,107,83,50,26,99,62)(11,108,84,51,27,100,63)(12,109,85,52,28,101,64)(13,110,86,53,29,102,57)(14,111,87,54,30,103,58)(15,112,88,55,31,104,59)(16,105,81,56,32,97,60), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,34)(35,40)(36,39)(37,38)(41,42)(43,48)(44,47)(45,46)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95)(97,100)(98,99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(98,102)(100,104)(106,110)(108,112), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(49,68)(50,69)(51,70)(52,71)(53,72)(54,65)(55,66)(56,67)(57,80)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,89)(88,90) );
G=PermutationGroup([(1,75,42,18,91,67,34),(2,76,43,19,92,68,35),(3,77,44,20,93,69,36),(4,78,45,21,94,70,37),(5,79,46,22,95,71,38),(6,80,47,23,96,72,39),(7,73,48,24,89,65,40),(8,74,41,17,90,66,33),(9,106,82,49,25,98,61),(10,107,83,50,26,99,62),(11,108,84,51,27,100,63),(12,109,85,52,28,101,64),(13,110,86,53,29,102,57),(14,111,87,54,30,103,58),(15,112,88,55,31,104,59),(16,105,81,56,32,97,60)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,26),(27,32),(28,31),(29,30),(33,34),(35,40),(36,39),(37,38),(41,42),(43,48),(44,47),(45,46),(49,50),(51,56),(52,55),(53,54),(57,58),(59,64),(60,63),(61,62),(65,68),(66,67),(69,72),(70,71),(73,76),(74,75),(77,80),(78,79),(81,84),(82,83),(85,88),(86,87),(89,92),(90,91),(93,96),(94,95),(97,100),(98,99),(101,104),(102,103),(105,108),(106,107),(109,112),(110,111)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(66,70),(68,72),(74,78),(76,80),(82,86),(84,88),(90,94),(92,96),(98,102),(100,104),(106,110),(108,112)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,41),(16,42),(17,112),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34),(49,68),(50,69),(51,70),(52,71),(53,72),(54,65),(55,66),(56,67),(57,80),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,89),(88,90)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 20 | 0 | 111 | 0 |
| 20 | 0 | 112 | 112 |
| 30 | 1 | 93 | 0 |
| 30 | 0 | 93 | 0 |
| 20 | 0 | 111 | 0 |
| 0 | 0 | 112 | 1 |
| 30 | 0 | 93 | 0 |
| 30 | 1 | 93 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 20 | 0 | 112 | 0 |
| 20 | 0 | 0 | 112 |
| 15 | 83 | 0 | 0 |
| 15 | 98 | 0 | 0 |
| 74 | 39 | 0 | 98 |
| 0 | 39 | 15 | 0 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[20,20,30,30,0,0,1,0,111,112,93,93,0,112,0,0],[20,0,30,30,0,0,0,1,111,112,93,93,0,1,0,0],[1,0,20,20,0,1,0,0,0,0,112,0,0,0,0,112],[15,15,74,0,83,98,39,39,0,0,0,15,0,0,98,0] >;
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14X | 14Y | ··· | 14AV | 28A | ··· | 28L | 28M | ··· | 28AD | 28AE | ··· | 28BB | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C7×D4 | C7×D4 | D8⋊C22 | C7×D8⋊C22 |
| kernel | C7×D8⋊C22 | C14×M4(2) | C7×C4○D8 | C7×C8⋊C22 | C7×C8.C22 | C14×C4○D4 | D8⋊C22 | C2×M4(2) | C4○D8 | C8⋊C22 | C8.C22 | C2×C4○D4 | C2×C28 | C22×C14 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 6 | 6 | 24 | 24 | 24 | 12 | 3 | 1 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_7\times D_8\rtimes C_2^2
% in TeX
G:=Group("C7xD8:C2^2"); // GroupNames label
G:=SmallGroup(448,1358);
// by ID
G=gap.SmallGroup(448,1358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,808,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^8=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d=b^5,b*e=e*b,d*c*d=e*c*e=b^4*c,d*e=e*d>;
// generators/relations