direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22.2D28, C23.15D28, C24.11D14, (C23×D7)⋊4C4, C14⋊1(C23⋊C4), C22⋊C4⋊36D14, C23.17(C4×D7), (C22×Dic7)⋊5C4, (C22×C14).66D4, C22.14(C2×D28), C23.74(C7⋊D4), C23.D7⋊42C22, C23.72(C22×D7), C22.44(D14⋊C4), (C23×C14).37C22, (C22×C14).111C23, C7⋊2(C2×C23⋊C4), (C2×C7⋊D4)⋊3C4, C2.8(C2×D14⋊C4), (C2×C22⋊C4)⋊1D7, C22.18(C2×C4×D7), (C14×C22⋊C4)⋊1C2, (C2×Dic7)⋊2(C2×C4), (C22×D7)⋊2(C2×C4), (C2×C23.D7)⋊1C2, (C2×C14).433(C2×D4), C14.35(C2×C22⋊C4), (C22×C7⋊D4).1C2, C22.26(C2×C7⋊D4), (C7×C22⋊C4)⋊44C22, (C2×C14).12(C22×C4), (C22×C14).52(C2×C4), (C2×C7⋊D4).83C22, (C2×C14).56(C22⋊C4), SmallGroup(448,488)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22.2D28
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 1172 in 210 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×6], C22 [×7], C22 [×18], C7, C2×C4 [×12], D4 [×8], C23 [×7], C23 [×8], D7 [×2], C14, C14 [×2], C14 [×6], C22⋊C4 [×2], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×8], C24, C24, Dic7 [×4], C28 [×2], D14 [×8], C2×C14 [×7], C2×C14 [×10], C23⋊C4 [×4], C2×C22⋊C4, C2×C22⋊C4, C22×D4, C2×Dic7 [×2], C2×Dic7 [×6], C7⋊D4 [×8], C2×C28 [×4], C22×D7 [×2], C22×D7 [×4], C22×C14 [×7], C22×C14 [×2], C2×C23⋊C4, C23.D7 [×2], C23.D7, C7×C22⋊C4 [×2], C7×C22⋊C4, C22×Dic7, C22×Dic7, C2×C7⋊D4 [×4], C2×C7⋊D4 [×4], C22×C28, C23×D7, C23×C14, C22.2D28 [×4], C2×C23.D7, C14×C22⋊C4, C22×C7⋊D4, C2×C22.2D28
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D14 [×3], C23⋊C4 [×2], C2×C22⋊C4, C4×D7 [×2], D28 [×2], C7⋊D4 [×2], C22×D7, C2×C23⋊C4, D14⋊C4 [×4], C2×C4×D7, C2×D28, C2×C7⋊D4, C22.2D28 [×2], C2×D14⋊C4, C2×C22.2D28
►On 112 points
Generators in S112
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 109)(30 110)(31 111)(32 112)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)
(1 102)(3 104)(5 106)(7 108)(9 110)(11 112)(13 86)(15 88)(17 90)(19 92)(21 94)(23 96)(25 98)(27 100)(30 82)(32 84)(34 58)(36 60)(38 62)(40 64)(42 66)(44 68)(46 70)(48 72)(50 74)(52 76)(54 78)(56 80)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 81)(30 82)(31 83)(32 84)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)
(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 56 102 80)(2 55)(3 78 104 54)(4 77)(5 52 106 76)(6 51)(7 74 108 50)(8 73)(9 48 110 72)(10 47)(11 70 112 46)(12 69)(13 44 86 68)(14 43)(15 66 88 42)(16 65)(17 40 90 64)(18 39)(19 62 92 38)(20 61)(21 36 94 60)(22 35)(23 58 96 34)(24 57)(25 32 98 84)(26 31)(27 82 100 30)(28 81)(29 101)(33 97)(37 93)(41 89)(45 85)(49 109)(53 105)(59 95)(63 91)(67 87)(71 111)(75 107)(79 103)(83 99)
G:=sub<Sym(112)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,109)(30,110)(31,111)(32,112)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108), (1,102)(3,104)(5,106)(7,108)(9,110)(11,112)(13,86)(15,88)(17,90)(19,92)(21,94)(23,96)(25,98)(27,100)(30,82)(32,84)(34,58)(36,60)(38,62)(40,64)(42,66)(44,68)(46,70)(48,72)(50,74)(52,76)(54,78)(56,80), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,81)(30,82)(31,83)(32,84)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (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), (1,56,102,80)(2,55)(3,78,104,54)(4,77)(5,52,106,76)(6,51)(7,74,108,50)(8,73)(9,48,110,72)(10,47)(11,70,112,46)(12,69)(13,44,86,68)(14,43)(15,66,88,42)(16,65)(17,40,90,64)(18,39)(19,62,92,38)(20,61)(21,36,94,60)(22,35)(23,58,96,34)(24,57)(25,32,98,84)(26,31)(27,82,100,30)(28,81)(29,101)(33,97)(37,93)(41,89)(45,85)(49,109)(53,105)(59,95)(63,91)(67,87)(71,111)(75,107)(79,103)(83,99)>;
G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,109)(30,110)(31,111)(32,112)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108), (1,102)(3,104)(5,106)(7,108)(9,110)(11,112)(13,86)(15,88)(17,90)(19,92)(21,94)(23,96)(25,98)(27,100)(30,82)(32,84)(34,58)(36,60)(38,62)(40,64)(42,66)(44,68)(46,70)(48,72)(50,74)(52,76)(54,78)(56,80), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,81)(30,82)(31,83)(32,84)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (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), (1,56,102,80)(2,55)(3,78,104,54)(4,77)(5,52,106,76)(6,51)(7,74,108,50)(8,73)(9,48,110,72)(10,47)(11,70,112,46)(12,69)(13,44,86,68)(14,43)(15,66,88,42)(16,65)(17,40,90,64)(18,39)(19,62,92,38)(20,61)(21,36,94,60)(22,35)(23,58,96,34)(24,57)(25,32,98,84)(26,31)(27,82,100,30)(28,81)(29,101)(33,97)(37,93)(41,89)(45,85)(49,109)(53,105)(59,95)(63,91)(67,87)(71,111)(75,107)(79,103)(83,99) );
G=PermutationGroup([(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,109),(30,110),(31,111),(32,112),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108)], [(1,102),(3,104),(5,106),(7,108),(9,110),(11,112),(13,86),(15,88),(17,90),(19,92),(21,94),(23,96),(25,98),(27,100),(30,82),(32,84),(34,58),(36,60),(38,62),(40,64),(42,66),(44,68),(46,70),(48,72),(50,74),(52,76),(54,78),(56,80)], [(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,85),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,97),(25,98),(26,99),(27,100),(28,101),(29,81),(30,82),(31,83),(32,84),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80)], [(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)], [(1,56,102,80),(2,55),(3,78,104,54),(4,77),(5,52,106,76),(6,51),(7,74,108,50),(8,73),(9,48,110,72),(10,47),(11,70,112,46),(12,69),(13,44,86,68),(14,43),(15,66,88,42),(16,65),(17,40,90,64),(18,39),(19,62,92,38),(20,61),(21,36,94,60),(22,35),(23,58,96,34),(24,57),(25,32,98,84),(26,31),(27,82,100,30),(28,81),(29,101),(33,97),(37,93),(41,89),(45,85),(49,109),(53,105),(59,95),(63,91),(67,87),(71,111),(75,107),(79,103),(83,99)])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | C23⋊C4 | C22.2D28 |
| kernel | C2×C22.2D28 | C22.2D28 | C2×C23.D7 | C14×C22⋊C4 | C22×C7⋊D4 | C22×Dic7 | C2×C7⋊D4 | C23×D7 | C22×C14 | C2×C22⋊C4 | C22⋊C4 | C24 | C23 | C23 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 3 | 6 | 3 | 12 | 12 | 12 | 2 | 12 |
Matrix representation of C2×C22.2D28 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 27 | 17 | 1 | 0 |
| 0 | 0 | 12 | 20 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 27 | 6 |
| 0 | 0 | 15 | 22 | 23 | 16 |
| 0 | 0 | 24 | 2 | 20 | 15 |
| 0 | 0 | 27 | 1 | 14 | 7 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 27 | 0 | 0 |
| 0 | 0 | 13 | 24 | 0 | 0 |
| 0 | 0 | 6 | 19 | 1 | 26 |
| 0 | 0 | 8 | 23 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,27,12,0,0,0,28,17,20,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,9,15,24,27,0,0,14,22,2,1,0,0,27,23,20,14,0,0,6,16,15,7],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,5,13,6,8,0,0,27,24,19,23,0,0,0,0,1,0,0,0,0,0,26,28] >;
C2×C22.2D28 in GAP, Magma, Sage, TeX
C_2\times C_2^2._2D_{28} % in TeX
G:=Group("C2xC2^2.2D28"); // GroupNames label
G:=SmallGroup(448,488);
// by ID
G=gap.SmallGroup(448,488);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,1123,438,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations