metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊22D14, C14.1292+ (1+4), (C2×Q8)⋊10D14, (C4×C28)⋊29C22, D14⋊C4⋊6C22, C22⋊C4⋊21D14, C4.4D4⋊16D7, C23⋊D14⋊25C2, C22⋊D28⋊26C2, (C2×D4).112D14, C4.D28⋊30C2, (C2×C28).83C23, (Q8×C14)⋊16C22, C28.23D4⋊24C2, (C2×C14).227C24, C7⋊2(C24⋊C22), (C4×Dic7)⋊37C22, (C2×D28).34C22, (C23×D7)⋊12C22, C23.D7⋊35C22, C2.77(D4⋊6D14), C2.53(D4⋊8D14), C23.49(C22×D7), Dic7.D4⋊43C2, (C2×Dic14)⋊10C22, (D4×C14).212C22, (C22×C14).57C23, (C22×D7).99C23, C22.248(C23×D7), (C2×Dic7).117C23, (C7×C4.4D4)⋊19C2, (C7×C22⋊C4)⋊32C22, (C2×C4).200(C22×D7), (C2×C7⋊D4).65C22, SmallGroup(448,1136)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1676 in 260 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×9], C22, C22 [×26], C7, C2×C4, C2×C4 [×4], C2×C4 [×4], D4 [×9], Q8 [×3], C23 [×2], C23 [×10], D7 [×4], C14, C14 [×2], C14 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×14], C2×D4, C2×D4 [×8], C2×Q8, C2×Q8 [×2], C24 [×2], Dic7 [×4], C28 [×5], D14 [×20], C2×C14, C2×C14 [×6], C22≀C2 [×6], C4.4D4, C4.4D4 [×8], Dic14 [×2], D28 [×4], C2×Dic7 [×4], C7⋊D4 [×4], C2×C28, C2×C28 [×4], C7×D4, C7×Q8, C22×D7 [×4], C22×D7 [×6], C22×C14 [×2], C24⋊C22, C4×Dic7 [×2], D14⋊C4 [×12], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×4], C2×Dic14 [×2], C2×D28 [×4], C2×C7⋊D4 [×4], D4×C14, Q8×C14, C23×D7 [×2], C4.D28 [×2], C22⋊D28 [×4], Dic7.D4 [×4], C23⋊D14 [×2], C28.23D4 [×2], C7×C4.4D4, C42⋊22D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×3], C22×D7 [×7], C24⋊C22, C23×D7, D4⋊6D14, D4⋊8D14 [×2], C42⋊22D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >
►On 112 points
Generators in S112
(1 8 74 81)(2 82 75 9)(3 10 76 83)(4 84 77 11)(5 12 78 71)(6 72 79 13)(7 14 80 73)(15 62 101 37)(16 38 102 63)(17 64 103 39)(18 40 104 65)(19 66 105 41)(20 42 106 67)(21 68 107 29)(22 30 108 69)(23 70 109 31)(24 32 110 57)(25 58 111 33)(26 34 112 59)(27 60 99 35)(28 36 100 61)(43 50 92 85)(44 86 93 51)(45 52 94 87)(46 88 95 53)(47 54 96 89)(48 90 97 55)(49 56 98 91)
(1 111 49 65)(2 41 50 26)(3 99 51 67)(4 29 52 28)(5 101 53 69)(6 31 54 16)(7 103 55 57)(8 33 56 18)(9 105 43 59)(10 35 44 20)(11 107 45 61)(12 37 46 22)(13 109 47 63)(14 39 48 24)(15 88 30 78)(17 90 32 80)(19 92 34 82)(21 94 36 84)(23 96 38 72)(25 98 40 74)(27 86 42 76)(58 91 104 81)(60 93 106 83)(62 95 108 71)(64 97 110 73)(66 85 112 75)(68 87 100 77)(70 89 102 79)
(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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 98)(25 97)(26 96)(27 95)(28 94)(29 84)(30 83)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 72)(42 71)(43 102)(44 101)(45 100)(46 99)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 104)(56 103)
G:=sub<Sym(112)| (1,8,74,81)(2,82,75,9)(3,10,76,83)(4,84,77,11)(5,12,78,71)(6,72,79,13)(7,14,80,73)(15,62,101,37)(16,38,102,63)(17,64,103,39)(18,40,104,65)(19,66,105,41)(20,42,106,67)(21,68,107,29)(22,30,108,69)(23,70,109,31)(24,32,110,57)(25,58,111,33)(26,34,112,59)(27,60,99,35)(28,36,100,61)(43,50,92,85)(44,86,93,51)(45,52,94,87)(46,88,95,53)(47,54,96,89)(48,90,97,55)(49,56,98,91), (1,111,49,65)(2,41,50,26)(3,99,51,67)(4,29,52,28)(5,101,53,69)(6,31,54,16)(7,103,55,57)(8,33,56,18)(9,105,43,59)(10,35,44,20)(11,107,45,61)(12,37,46,22)(13,109,47,63)(14,39,48,24)(15,88,30,78)(17,90,32,80)(19,92,34,82)(21,94,36,84)(23,96,38,72)(25,98,40,74)(27,86,42,76)(58,91,104,81)(60,93,106,83)(62,95,108,71)(64,97,110,73)(66,85,112,75)(68,87,100,77)(70,89,102,79), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,98)(25,97)(26,96)(27,95)(28,94)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,102)(44,101)(45,100)(46,99)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)>;
G:=Group( (1,8,74,81)(2,82,75,9)(3,10,76,83)(4,84,77,11)(5,12,78,71)(6,72,79,13)(7,14,80,73)(15,62,101,37)(16,38,102,63)(17,64,103,39)(18,40,104,65)(19,66,105,41)(20,42,106,67)(21,68,107,29)(22,30,108,69)(23,70,109,31)(24,32,110,57)(25,58,111,33)(26,34,112,59)(27,60,99,35)(28,36,100,61)(43,50,92,85)(44,86,93,51)(45,52,94,87)(46,88,95,53)(47,54,96,89)(48,90,97,55)(49,56,98,91), (1,111,49,65)(2,41,50,26)(3,99,51,67)(4,29,52,28)(5,101,53,69)(6,31,54,16)(7,103,55,57)(8,33,56,18)(9,105,43,59)(10,35,44,20)(11,107,45,61)(12,37,46,22)(13,109,47,63)(14,39,48,24)(15,88,30,78)(17,90,32,80)(19,92,34,82)(21,94,36,84)(23,96,38,72)(25,98,40,74)(27,86,42,76)(58,91,104,81)(60,93,106,83)(62,95,108,71)(64,97,110,73)(66,85,112,75)(68,87,100,77)(70,89,102,79), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,98)(25,97)(26,96)(27,95)(28,94)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,102)(44,101)(45,100)(46,99)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103) );
G=PermutationGroup([(1,8,74,81),(2,82,75,9),(3,10,76,83),(4,84,77,11),(5,12,78,71),(6,72,79,13),(7,14,80,73),(15,62,101,37),(16,38,102,63),(17,64,103,39),(18,40,104,65),(19,66,105,41),(20,42,106,67),(21,68,107,29),(22,30,108,69),(23,70,109,31),(24,32,110,57),(25,58,111,33),(26,34,112,59),(27,60,99,35),(28,36,100,61),(43,50,92,85),(44,86,93,51),(45,52,94,87),(46,88,95,53),(47,54,96,89),(48,90,97,55),(49,56,98,91)], [(1,111,49,65),(2,41,50,26),(3,99,51,67),(4,29,52,28),(5,101,53,69),(6,31,54,16),(7,103,55,57),(8,33,56,18),(9,105,43,59),(10,35,44,20),(11,107,45,61),(12,37,46,22),(13,109,47,63),(14,39,48,24),(15,88,30,78),(17,90,32,80),(19,92,34,82),(21,94,36,84),(23,96,38,72),(25,98,40,74),(27,86,42,76),(58,91,104,81),(60,93,106,83),(62,95,108,71),(64,97,110,73),(66,85,112,75),(68,87,100,77),(70,89,102,79)], [(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)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,93),(16,92),(17,91),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,98),(25,97),(26,96),(27,95),(28,94),(29,84),(30,83),(31,82),(32,81),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(41,72),(42,71),(43,102),(44,101),(45,100),(46,99),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,104),(56,103)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 1 | 0 | 9 | 10 | 0 | 0 | 0 | 0 |
| 0 | 1 | 21 | 26 | 0 | 0 | 0 | 0 |
| 22 | 25 | 28 | 0 | 0 | 0 | 0 | 0 |
| 9 | 21 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 | 4 | 14 |
| 0 | 0 | 0 | 0 | 21 | 28 | 1 | 25 |
| 27 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 27 | 0 |
| 0 | 0 | 0 | 0 | 21 | 19 | 0 | 27 |
| 0 | 0 | 0 | 0 | 20 | 28 | 0 | 23 |
| 0 | 0 | 0 | 0 | 11 | 12 | 8 | 10 |
| 21 | 4 | 12 | 24 | 0 | 0 | 0 | 0 |
| 5 | 1 | 8 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 7 | 14 | 8 |
| 0 | 0 | 0 | 0 | 22 | 21 | 13 | 26 |
| 24 | 7 | 15 | 16 | 0 | 0 | 0 | 0 |
| 9 | 5 | 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 17 | 5 | 18 |
| 0 | 0 | 0 | 0 | 9 | 4 | 0 | 24 |
| 0 | 0 | 0 | 0 | 4 | 22 | 0 | 14 |
| 0 | 0 | 0 | 0 | 15 | 22 | 9 | 11 |
G:=sub<GL(8,GF(29))| [1,0,22,9,0,0,0,0,0,1,25,21,0,0,0,0,9,21,28,0,0,0,0,0,10,26,0,28,0,0,0,0,0,0,0,0,4,1,1,21,0,0,0,0,14,25,17,28,0,0,0,0,0,0,4,1,0,0,0,0,0,0,14,25],[27,28,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,8,16,0,0,0,0,0,0,5,21,0,0,0,0,0,0,0,0,0,21,20,11,0,0,0,0,6,19,28,12,0,0,0,0,27,0,0,8,0,0,0,0,0,27,23,10],[21,5,0,0,0,0,0,0,4,1,0,0,0,0,0,0,12,8,0,22,0,0,0,0,24,18,25,7,0,0,0,0,0,0,0,0,15,16,22,22,0,0,0,0,21,3,7,21,0,0,0,0,0,0,14,13,0,0,0,0,0,0,8,26],[24,9,0,0,0,0,0,0,7,5,0,0,0,0,0,0,15,12,25,5,0,0,0,0,16,17,26,4,0,0,0,0,0,0,0,0,14,9,4,15,0,0,0,0,17,4,22,22,0,0,0,0,5,0,0,9,0,0,0,0,18,24,14,11] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D4⋊8D14 |
| kernel | C42⋊22D14 | C4.D28 | C22⋊D28 | Dic7.D4 | C23⋊D14 | C28.23D4 | C7×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 3 | 3 | 12 | 3 | 3 | 3 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{22}D_{14} % in TeX
G:=Group("C4^2:22D14"); // GroupNames label
G:=SmallGroup(448,1136);
// by ID
G=gap.SmallGroup(448,1136);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,1571,570,297,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀