direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D7×SD16, C56⋊6C23, C28.5C24, D28.3C23, Dic14⋊2C23, C7⋊C8⋊7C23, (C2×C8)⋊28D14, C4.42(D4×D7), C8⋊6(C22×D7), (C2×Q8)⋊20D14, (C4×D7).28D4, C14⋊2(C2×SD16), C28.80(C2×D4), Q8⋊D7⋊7C22, Q8⋊1(C22×D7), (Q8×D7)⋊5C22, (C7×Q8)⋊1C23, C4.5(C23×D7), C7⋊2(C22×SD16), (C2×C56)⋊18C22, D14.64(C2×D4), (C8×D7)⋊17C22, D4.D7⋊9C22, (C7×D4).3C23, (D4×D7).5C22, D4.3(C22×D7), (C14×SD16)⋊10C2, (C2×D4).181D14, C56⋊C2⋊17C22, Dic7.12(C2×D4), (Q8×C14)⋊17C22, (C4×D7).25C23, C22.138(D4×D7), (C2×C28).522C23, (C2×Dic7).122D4, (C7×SD16)⋊12C22, (C22×D7).111D4, C14.106(C22×D4), (C2×Dic14)⋊37C22, (D4×C14).163C22, (C2×D28).177C22, (D7×C2×C8)⋊9C2, (C2×Q8×D7)⋊14C2, C2.79(C2×D4×D7), (C2×D4×D7).10C2, (C2×C7⋊C8)⋊36C22, (C2×Q8⋊D7)⋊25C2, (C2×C56⋊C2)⋊31C2, (C2×D4.D7)⋊27C2, (C2×C14).395(C2×D4), (C2×C4×D7).257C22, (C2×C4).611(C22×D7), SmallGroup(448,1211)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1668 in 298 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×6], C22, C22 [×22], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×8], Q8 [×2], Q8 [×8], C23 [×11], D7 [×4], D7 [×2], C14, C14 [×2], C14 [×2], C2×C8, C2×C8 [×5], SD16 [×4], SD16 [×12], C22×C4 [×2], C2×D4, C2×D4 [×8], C2×Q8, C2×Q8 [×8], C24, Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×6], D14 [×12], C2×C14, C2×C14 [×4], C22×C8, C2×SD16, C2×SD16 [×11], C22×D4, C22×Q8, C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14 [×5], C4×D7 [×4], C4×D7 [×4], D28 [×2], D28, C2×Dic7, C2×Dic7, C7⋊D4 [×4], C2×C28, C2×C28, C7×D4 [×2], C7×D4, C7×Q8 [×2], C7×Q8, C22×D7, C22×D7 [×9], C22×C14, C22×SD16, C8×D7 [×4], C56⋊C2 [×4], C2×C7⋊C8, D4.D7 [×4], Q8⋊D7 [×4], C2×C56, C7×SD16 [×4], C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, D4×D7 [×4], D4×D7 [×2], Q8×D7 [×4], Q8×D7 [×2], C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, D7×C2×C8, C2×C56⋊C2, D7×SD16 [×8], C2×D4.D7, C2×Q8⋊D7, C14×SD16, C2×D4×D7, C2×Q8×D7, C2×D7×SD16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, SD16 [×4], C2×D4 [×6], C24, D14 [×7], C2×SD16 [×6], C22×D4, C22×D7 [×7], C22×SD16, D4×D7 [×2], C23×D7, D7×SD16 [×2], C2×D4×D7, C2×D7×SD16
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=d3 >
►On 112 points
(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)(33 109)(34 110)(35 111)(36 112)(37 105)(38 106)(39 107)(40 108)(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)(73 100)(74 101)(75 102)(76 103)(77 104)(78 97)(79 98)(80 99)
(1 102 105 16 19 47 50)(2 103 106 9 20 48 51)(3 104 107 10 21 41 52)(4 97 108 11 22 42 53)(5 98 109 12 23 43 54)(6 99 110 13 24 44 55)(7 100 111 14 17 45 56)(8 101 112 15 18 46 49)(25 76 38 61 72 88 91)(26 77 39 62 65 81 92)(27 78 40 63 66 82 93)(28 79 33 64 67 83 94)(29 80 34 57 68 84 95)(30 73 35 58 69 85 96)(31 74 36 59 70 86 89)(32 75 37 60 71 87 90)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 13)(10 14)(11 15)(12 16)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 105)(24 106)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 71)(34 72)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 100)(42 101)(43 102)(44 103)(45 104)(46 97)(47 98)(48 99)(57 61)(58 62)(59 63)(60 64)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)
(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 5)(2 8)(4 6)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(49 51)(50 54)(53 55)(57 63)(59 61)(60 64)(66 68)(67 71)(70 72)(74 76)(75 79)(78 80)(82 84)(83 87)(86 88)(89 91)(90 94)(93 95)(97 99)(98 102)(101 103)(105 109)(106 112)(108 110)
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)(33,109)(34,110)(35,111)(36,112)(37,105)(38,106)(39,107)(40,108)(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)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99), (1,102,105,16,19,47,50)(2,103,106,9,20,48,51)(3,104,107,10,21,41,52)(4,97,108,11,22,42,53)(5,98,109,12,23,43,54)(6,99,110,13,24,44,55)(7,100,111,14,17,45,56)(8,101,112,15,18,46,49)(25,76,38,61,72,88,91)(26,77,39,62,65,81,92)(27,78,40,63,66,82,93)(28,79,33,64,67,83,94)(29,80,34,57,68,84,95)(30,73,35,58,69,85,96)(31,74,36,59,70,86,89)(32,75,37,60,71,87,90), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,13)(10,14)(11,15)(12,16)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,100)(42,101)(43,102)(44,103)(45,104)(46,97)(47,98)(48,99)(57,61)(58,62)(59,63)(60,64)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88), (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), (1,5)(2,8)(4,6)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,51)(50,54)(53,55)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,91)(90,94)(93,95)(97,99)(98,102)(101,103)(105,109)(106,112)(108,110)>;
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)(33,109)(34,110)(35,111)(36,112)(37,105)(38,106)(39,107)(40,108)(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)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99), (1,102,105,16,19,47,50)(2,103,106,9,20,48,51)(3,104,107,10,21,41,52)(4,97,108,11,22,42,53)(5,98,109,12,23,43,54)(6,99,110,13,24,44,55)(7,100,111,14,17,45,56)(8,101,112,15,18,46,49)(25,76,38,61,72,88,91)(26,77,39,62,65,81,92)(27,78,40,63,66,82,93)(28,79,33,64,67,83,94)(29,80,34,57,68,84,95)(30,73,35,58,69,85,96)(31,74,36,59,70,86,89)(32,75,37,60,71,87,90), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,13)(10,14)(11,15)(12,16)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,100)(42,101)(43,102)(44,103)(45,104)(46,97)(47,98)(48,99)(57,61)(58,62)(59,63)(60,64)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88), (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), (1,5)(2,8)(4,6)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,51)(50,54)(53,55)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,91)(90,94)(93,95)(97,99)(98,102)(101,103)(105,109)(106,112)(108,110) );
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),(33,109),(34,110),(35,111),(36,112),(37,105),(38,106),(39,107),(40,108),(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),(73,100),(74,101),(75,102),(76,103),(77,104),(78,97),(79,98),(80,99)], [(1,102,105,16,19,47,50),(2,103,106,9,20,48,51),(3,104,107,10,21,41,52),(4,97,108,11,22,42,53),(5,98,109,12,23,43,54),(6,99,110,13,24,44,55),(7,100,111,14,17,45,56),(8,101,112,15,18,46,49),(25,76,38,61,72,88,91),(26,77,39,62,65,81,92),(27,78,40,63,66,82,93),(28,79,33,64,67,83,94),(29,80,34,57,68,84,95),(30,73,35,58,69,85,96),(31,74,36,59,70,86,89),(32,75,37,60,71,87,90)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,13),(10,14),(11,15),(12,16),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,105),(24,106),(25,95),(26,96),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,71),(34,72),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,100),(42,101),(43,102),(44,103),(45,104),(46,97),(47,98),(48,99),(57,61),(58,62),(59,63),(60,64),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88)], [(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)], [(1,5),(2,8),(4,6),(9,15),(11,13),(12,16),(18,20),(19,23),(22,24),(25,31),(27,29),(28,32),(33,37),(34,40),(36,38),(42,44),(43,47),(46,48),(49,51),(50,54),(53,55),(57,63),(59,61),(60,64),(66,68),(67,71),(70,72),(74,76),(75,79),(78,80),(82,84),(83,87),(86,88),(89,91),(90,94),(93,95),(97,99),(98,102),(101,103),(105,109),(106,112),(108,110)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 1 | 0 | 0 |
| 112 | 79 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 100 | 100 |
| 0 | 0 | 13 | 100 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,100,13,0,0,100,100],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,1] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 7 | 7 | 7 | 7 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | D14 | D14 | D4×D7 | D4×D7 | D7×SD16 |
| kernel | C2×D7×SD16 | D7×C2×C8 | C2×C56⋊C2 | D7×SD16 | C2×D4.D7 | C2×Q8⋊D7 | C14×SD16 | C2×D4×D7 | C2×Q8×D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×SD16 | D14 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 8 | 3 | 12 | 3 | 3 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_7\times SD_{16} % in TeX
G:=Group("C2xD7xSD16"); // GroupNames label
G:=SmallGroup(448,1211);
// by ID
G=gap.SmallGroup(448,1211);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,185,136,438,235,102,18822]);
// 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^3>;
// generators/relations