direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22⋊D28, C23⋊5D28, C24.54D14, C14⋊1C22≀C2, D14⋊10(C2×D4), (C2×C28)⋊3C23, (D7×C24)⋊1C2, C22⋊3(C2×D28), (C22×C14)⋊9D4, (C22×C4)⋊7D14, C22⋊C4⋊37D14, (C22×D7)⋊13D4, (C22×D28)⋊5C2, C14.6(C22×D4), C2.8(C22×D28), D14⋊C4⋊44C22, (C2×D28)⋊43C22, (C2×C14).31C24, (C22×C28)⋊7C22, (C2×Dic7)⋊1C23, C22.125(D4×D7), (C22×D7)⋊1C23, (C23×D7)⋊3C22, C22.70(C23×D7), (C23×C14).57C22, (C22×Dic7)⋊6C22, C23.146(C22×D7), (C22×C14).123C23, C2.8(C2×D4×D7), C7⋊1(C2×C22≀C2), (C2×C14)⋊4(C2×D4), (C2×C4)⋊3(C22×D7), (C2×D14⋊C4)⋊16C2, (C2×C22⋊C4)⋊10D7, (C22×C7⋊D4)⋊3C2, (C14×C22⋊C4)⋊13C2, (C2×C7⋊D4)⋊34C22, (C7×C22⋊C4)⋊46C22, SmallGroup(448,940)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22⋊D28
G = < a,b,c,d,e | a2=b2=c2=d28=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 3988 in 662 conjugacy classes, 143 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C22×D4, C25, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C22≀C2, D14⋊C4, C7×C22⋊C4, C2×D28, C2×D28, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×D7, C23×D7, C23×C14, C22⋊D28, C2×D14⋊C4, C14×C22⋊C4, C22×D28, C22×C7⋊D4, D7×C24, C2×C22⋊D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22≀C2, C22×D4, D28, C22×D7, C2×C22≀C2, C2×D28, D4×D7, C23×D7, C22⋊D28, C22×D28, C2×D4×D7, C2×C22⋊D28
►On 112 points
Generators in S112
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)
(1 91)(2 80)(3 93)(4 82)(5 95)(6 84)(7 97)(8 58)(9 99)(10 60)(11 101)(12 62)(13 103)(14 64)(15 105)(16 66)(17 107)(18 68)(19 109)(20 70)(21 111)(22 72)(23 85)(24 74)(25 87)(26 76)(27 89)(28 78)(29 67)(30 108)(31 69)(32 110)(33 71)(34 112)(35 73)(36 86)(37 75)(38 88)(39 77)(40 90)(41 79)(42 92)(43 81)(44 94)(45 83)(46 96)(47 57)(48 98)(49 59)(50 100)(51 61)(52 102)(53 63)(54 104)(55 65)(56 106)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 102)(30 101)(31 100)(32 99)(33 98)(34 97)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 104)(56 103)
G:=sub<Sym(112)| (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66), (1,91)(2,80)(3,93)(4,82)(5,95)(6,84)(7,97)(8,58)(9,99)(10,60)(11,101)(12,62)(13,103)(14,64)(15,105)(16,66)(17,107)(18,68)(19,109)(20,70)(21,111)(22,72)(23,85)(24,74)(25,87)(26,76)(27,89)(28,78)(29,67)(30,108)(31,69)(32,110)(33,71)(34,112)(35,73)(36,86)(37,75)(38,88)(39,77)(40,90)(41,79)(42,92)(43,81)(44,94)(45,83)(46,96)(47,57)(48,98)(49,59)(50,100)(51,61)(52,102)(53,63)(54,104)(55,65)(56,106), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)>;
G:=Group( (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66), (1,91)(2,80)(3,93)(4,82)(5,95)(6,84)(7,97)(8,58)(9,99)(10,60)(11,101)(12,62)(13,103)(14,64)(15,105)(16,66)(17,107)(18,68)(19,109)(20,70)(21,111)(22,72)(23,85)(24,74)(25,87)(26,76)(27,89)(28,78)(29,67)(30,108)(31,69)(32,110)(33,71)(34,112)(35,73)(36,86)(37,75)(38,88)(39,77)(40,90)(41,79)(42,92)(43,81)(44,94)(45,83)(46,96)(47,57)(48,98)(49,59)(50,100)(51,61)(52,102)(53,63)(54,104)(55,65)(56,106), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103) );
G=PermutationGroup([[(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66)], [(1,91),(2,80),(3,93),(4,82),(5,95),(6,84),(7,97),(8,58),(9,99),(10,60),(11,101),(12,62),(13,103),(14,64),(15,105),(16,66),(17,107),(18,68),(19,109),(20,70),(21,111),(22,72),(23,85),(24,74),(25,87),(26,76),(27,89),(28,78),(29,67),(30,108),(31,69),(32,110),(33,71),(34,112),(35,73),(36,86),(37,75),(38,88),(39,77),(40,90),(41,79),(42,92),(43,81),(44,94),(45,83),(46,96),(47,57),(48,98),(49,59),(50,100),(51,61),(52,102),(53,63),(54,104),(55,65),(56,106)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,60),(20,59),(21,58),(22,57),(23,84),(24,83),(25,82),(26,81),(27,80),(28,79),(29,102),(30,101),(31,100),(32,99),(33,98),(34,97),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,104),(56,103)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 2T | 2U | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 28 | 28 | 4 | 4 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D4×D7 |
| kernel | C2×C22⋊D28 | C22⋊D28 | C2×D14⋊C4 | C14×C22⋊C4 | C22×D28 | C22×C7⋊D4 | D7×C24 | C22×D7 | C22×C14 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 8 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 12 | 6 | 3 | 24 | 12 |
Matrix representation of C2×C22⋊D28 ►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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 15 | 11 | 0 | 0 | 0 | 0 |
| 7 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 27 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 10 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[15,7,0,0,0,0,11,11,0,0,0,0,0,0,28,27,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[7,10,0,0,0,0,1,22,0,0,0,0,0,0,1,0,0,0,0,0,28,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2×C22⋊D28 in GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes D_{28} % in TeX
G:=Group("C2xC2^2:D28"); // GroupNames label
G:=SmallGroup(448,940);
// by ID
G=gap.SmallGroup(448,940);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,297,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations