direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D8⋊C22, C56.51C23, C28.84C24, 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, C4.7(C23×C14), C8.2(C22×C14), C23.20(C7×D4), (D4×C14)⋊67C22, M4(2)⋊5(C2×C14), (C2×M4(2))⋊5C14, (C7×Q16)⋊18C22, (Q8×C14)⋊56C22, (C7×D4).37C23, D4.4(C22×C14), C22.25(D4×C14), (C22×C14).38D4, (C7×Q8).38C23, Q8.4(C22×C14), (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
Generators and relations for C7×D8⋊C22
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 >
Subgroups: 402 in 262 conjugacy classes, 158 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, D8⋊C22, C2×C56, C7×M4(2), C7×D8, C7×SD16, C7×Q16, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C14×M4(2), C7×C4○D8, C7×C8⋊C22, C7×C8.C22, C14×C4○D4, C7×D8⋊C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C7×D4, C22×C14, D8⋊C22, D4×C14, C23×C14, D4×C2×C14, C7×D8⋊C22
►On 112 points
Generators in S112
(1 79 42 22 91 71 34)(2 80 43 23 92 72 35)(3 73 44 24 93 65 36)(4 74 45 17 94 66 37)(5 75 46 18 95 67 38)(6 76 47 19 96 68 39)(7 77 48 20 89 69 40)(8 78 41 21 90 70 33)(9 111 83 54 26 103 58)(10 112 84 55 27 104 59)(11 105 85 56 28 97 60)(12 106 86 49 29 98 61)(13 107 87 50 30 99 62)(14 108 88 51 31 100 63)(15 109 81 52 32 101 64)(16 110 82 53 25 102 57)
(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 16)(10 15)(11 14)(12 13)(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)(10 14)(12 16)(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 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 108)(18 109)(19 110)(20 111)(21 112)(22 105)(23 106)(24 107)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 76)(58 77)(59 78)(60 79)(61 80)(62 73)(63 74)(64 75)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
G:=sub<Sym(112)| (1,79,42,22,91,71,34)(2,80,43,23,92,72,35)(3,73,44,24,93,65,36)(4,74,45,17,94,66,37)(5,75,46,18,95,67,38)(6,76,47,19,96,68,39)(7,77,48,20,89,69,40)(8,78,41,21,90,70,33)(9,111,83,54,26,103,58)(10,112,84,55,27,104,59)(11,105,85,56,28,97,60)(12,106,86,49,29,98,61)(13,107,87,50,30,99,62)(14,108,88,51,31,100,63)(15,109,81,52,32,101,64)(16,110,82,53,25,102,57), (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)(9,16)(10,15)(11,14)(12,13)(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)(10,14)(12,16)(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)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,76)(58,77)(59,78)(60,79)(61,80)(62,73)(63,74)(64,75)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94)>;
G:=Group( (1,79,42,22,91,71,34)(2,80,43,23,92,72,35)(3,73,44,24,93,65,36)(4,74,45,17,94,66,37)(5,75,46,18,95,67,38)(6,76,47,19,96,68,39)(7,77,48,20,89,69,40)(8,78,41,21,90,70,33)(9,111,83,54,26,103,58)(10,112,84,55,27,104,59)(11,105,85,56,28,97,60)(12,106,86,49,29,98,61)(13,107,87,50,30,99,62)(14,108,88,51,31,100,63)(15,109,81,52,32,101,64)(16,110,82,53,25,102,57), (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)(9,16)(10,15)(11,14)(12,13)(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)(10,14)(12,16)(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)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,76)(58,77)(59,78)(60,79)(61,80)(62,73)(63,74)(64,75)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94) );
G=PermutationGroup([[(1,79,42,22,91,71,34),(2,80,43,23,92,72,35),(3,73,44,24,93,65,36),(4,74,45,17,94,66,37),(5,75,46,18,95,67,38),(6,76,47,19,96,68,39),(7,77,48,20,89,69,40),(8,78,41,21,90,70,33),(9,111,83,54,26,103,58),(10,112,84,55,27,104,59),(11,105,85,56,28,97,60),(12,106,86,49,29,98,61),(13,107,87,50,30,99,62),(14,108,88,51,31,100,63),(15,109,81,52,32,101,64),(16,110,82,53,25,102,57)], [(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),(9,16),(10,15),(11,14),(12,13),(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),(10,14),(12,16),(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),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,108),(18,109),(19,110),(20,111),(21,112),(22,105),(23,106),(24,107),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,72),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,76),(58,77),(59,78),(60,79),(61,80),(62,73),(63,74),(64,75),(81,95),(82,96),(83,89),(84,90),(85,91),(86,92),(87,93),(88,94)]])
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 |
Matrix representation of C7×D8⋊C22 ►in 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] >;
C7×D8⋊C22 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