direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D7×D8, C56⋊3C23, D28⋊1C23, C28.1C24, D56⋊15C22, C14⋊2(C2×D8), C7⋊C8⋊6C23, C7⋊2(C22×D8), (C14×D8)⋊7C2, (C2×C8)⋊26D14, C4.39(D4×D7), C8⋊5(C22×D7), (C2×D4)⋊27D14, (C2×D56)⋊19C2, D4⋊D7⋊8C22, (C4×D7).26D4, C28.76(C2×D4), (C7×D8)⋊9C22, (C7×D4)⋊1C23, D4⋊1(C22×D7), (D4×D7)⋊4C22, C4.1(C23×D7), (C2×C56)⋊11C22, D14.63(C2×D4), (C8×D7)⋊13C22, (D4×C14)⋊18C22, (C2×D28)⋊32C22, Dic7.11(C2×D4), (C4×D7).23C23, C22.135(D4×D7), (C2×C28).518C23, (C2×Dic7).121D4, (C22×D7).110D4, C14.102(C22×D4), (D7×C2×C8)⋊4C2, (C2×D4×D7)⋊21C2, C2.75(C2×D4×D7), (C2×D4⋊D7)⋊25C2, (C2×C7⋊C8)⋊35C22, (C2×C14).391(C2×D4), (C2×C4×D7).255C22, (C2×C4).608(C22×D7), SmallGroup(448,1207)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2212 in 338 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×2], C4 [×2], C22, C22 [×38], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×4], D4 [×16], C23 [×21], D7 [×4], D7 [×4], C14, C14 [×2], C14 [×4], C2×C8, C2×C8 [×5], D8 [×4], D8 [×12], C22×C4, C2×D4 [×2], C2×D4 [×16], C24 [×2], Dic7 [×2], C28 [×2], D14 [×6], D14 [×24], C2×C14, C2×C14 [×8], C22×C8, C2×D8, C2×D8 [×11], C22×D4 [×2], C7⋊C8 [×2], C56 [×2], C4×D7 [×4], D28 [×4], D28 [×2], C2×Dic7, C7⋊D4 [×8], C2×C28, C7×D4 [×4], C7×D4 [×2], C22×D7, C22×D7 [×18], C22×C14 [×2], C22×D8, C8×D7 [×4], D56 [×4], C2×C7⋊C8, D4⋊D7 [×8], C2×C56, C7×D8 [×4], C2×C4×D7, C2×D28 [×2], D4×D7 [×8], D4×D7 [×4], C2×C7⋊D4 [×2], D4×C14 [×2], C23×D7 [×2], D7×C2×C8, C2×D56, D7×D8 [×8], C2×D4⋊D7 [×2], C14×D8, C2×D4×D7 [×2], C2×D7×D8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, D8 [×4], C2×D4 [×6], C24, D14 [×7], C2×D8 [×6], C22×D4, C22×D7 [×7], C22×D8, D4×D7 [×2], C23×D7, D7×D8 [×2], C2×D4×D7, C2×D7×D8
Generators and relations
G = < a,b,c,d,e | a2=b7=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
►On 112 points
Generators in S112
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(33 102)(34 103)(35 104)(36 97)(37 98)(38 99)(39 100)(40 101)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 105)(80 106)
(1 105 17 100 82 57 54)(2 106 18 101 83 58 55)(3 107 19 102 84 59 56)(4 108 20 103 85 60 49)(5 109 21 104 86 61 50)(6 110 22 97 87 62 51)(7 111 23 98 88 63 52)(8 112 24 99 81 64 53)(9 90 28 75 65 35 46)(10 91 29 76 66 36 47)(11 92 30 77 67 37 48)(12 93 31 78 68 38 41)(13 94 32 79 69 39 42)(14 95 25 80 70 40 43)(15 96 26 73 71 33 44)(16 89 27 74 72 34 45)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 73)(16 74)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 81)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 40)(34 39)(35 38)(36 37)(41 46)(42 45)(43 44)(47 48)(49 54)(50 53)(51 52)(55 56)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(73 80)(74 79)(75 78)(76 77)(81 86)(82 85)(83 84)(87 88)(89 94)(90 93)(91 92)(95 96)(97 98)(99 104)(100 103)(101 102)(105 108)(106 107)(109 112)(110 111)
G:=sub<Sym(112)| (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(33,102)(34,103)(35,104)(36,97)(37,98)(38,99)(39,100)(40,101)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,105)(80,106), (1,105,17,100,82,57,54)(2,106,18,101,83,58,55)(3,107,19,102,84,59,56)(4,108,20,103,85,60,49)(5,109,21,104,86,61,50)(6,110,22,97,87,62,51)(7,111,23,98,88,63,52)(8,112,24,99,81,64,53)(9,90,28,75,65,35,46)(10,91,29,76,66,36,47)(11,92,30,77,67,37,48)(12,93,31,78,68,38,41)(13,94,32,79,69,39,42)(14,95,25,80,70,40,43)(15,96,26,73,71,33,44)(16,89,27,74,72,34,45), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,73)(16,74)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,40)(34,39)(35,38)(36,37)(41,46)(42,45)(43,44)(47,48)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(73,80)(74,79)(75,78)(76,77)(81,86)(82,85)(83,84)(87,88)(89,94)(90,93)(91,92)(95,96)(97,98)(99,104)(100,103)(101,102)(105,108)(106,107)(109,112)(110,111)>;
G:=Group( (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(33,102)(34,103)(35,104)(36,97)(37,98)(38,99)(39,100)(40,101)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,105)(80,106), (1,105,17,100,82,57,54)(2,106,18,101,83,58,55)(3,107,19,102,84,59,56)(4,108,20,103,85,60,49)(5,109,21,104,86,61,50)(6,110,22,97,87,62,51)(7,111,23,98,88,63,52)(8,112,24,99,81,64,53)(9,90,28,75,65,35,46)(10,91,29,76,66,36,47)(11,92,30,77,67,37,48)(12,93,31,78,68,38,41)(13,94,32,79,69,39,42)(14,95,25,80,70,40,43)(15,96,26,73,71,33,44)(16,89,27,74,72,34,45), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,73)(16,74)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,40)(34,39)(35,38)(36,37)(41,46)(42,45)(43,44)(47,48)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(73,80)(74,79)(75,78)(76,77)(81,86)(82,85)(83,84)(87,88)(89,94)(90,93)(91,92)(95,96)(97,98)(99,104)(100,103)(101,102)(105,108)(106,107)(109,112)(110,111) );
G=PermutationGroup([(1,32),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,69),(18,70),(19,71),(20,72),(21,65),(22,66),(23,67),(24,68),(33,102),(34,103),(35,104),(36,97),(37,98),(38,99),(39,100),(40,101),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,105),(80,106)], [(1,105,17,100,82,57,54),(2,106,18,101,83,58,55),(3,107,19,102,84,59,56),(4,108,20,103,85,60,49),(5,109,21,104,86,61,50),(6,110,22,97,87,62,51),(7,111,23,98,88,63,52),(8,112,24,99,81,64,53),(9,90,28,75,65,35,46),(10,91,29,76,66,36,47),(11,92,30,77,67,37,48),(12,93,31,78,68,38,41),(13,94,32,79,69,39,42),(14,95,25,80,70,40,43),(15,96,26,73,71,33,44),(16,89,27,74,72,34,45)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,73),(16,74),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,81),(25,95),(26,96),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(41,68),(42,69),(43,70),(44,71),(45,72),(46,65),(47,66),(48,67),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,26),(27,32),(28,31),(29,30),(33,40),(34,39),(35,38),(36,37),(41,46),(42,45),(43,44),(47,48),(49,54),(50,53),(51,52),(55,56),(57,60),(58,59),(61,64),(62,63),(65,68),(66,67),(69,72),(70,71),(73,80),(74,79),(75,78),(76,77),(81,86),(82,85),(83,84),(87,88),(89,94),(90,93),(91,92),(95,96),(97,98),(99,104),(100,103),(101,102),(105,108),(106,107),(109,112),(110,111)])
Matrix representation ►G ⊆ GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 88 | 1 | 0 | 0 |
| 0 | 111 | 104 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 34 | 79 | 0 | 0 |
| 0 | 24 | 79 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 |
| 0 | 0 | 0 | 31 | 62 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 |
| 0 | 0 | 0 | 31 | 0 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,88,111,0,0,0,1,104,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,24,0,0,0,79,79,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,31,0,0,0,51,62],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,31,0,0,0,62,0] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 28 | 28 | 28 | 28 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D8 | D14 | D14 | D14 | D4×D7 | D4×D7 | D7×D8 |
| kernel | C2×D7×D8 | D7×C2×C8 | C2×D56 | D7×D8 | C2×D4⋊D7 | C14×D8 | C2×D4×D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×D8 | D14 | C2×C8 | D8 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 8 | 3 | 12 | 6 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_7\times D_8
% in TeX
G:=Group("C2xD7xD8"); // GroupNames label
G:=SmallGroup(448,1207);
// by ID
G=gap.SmallGroup(448,1207);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀