metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊23D14, C14.1372+ (1+4), C4⋊C4⋊33D14, (C4×D28)⋊13C2, (C4×C28)⋊7C22, C28⋊1D4⋊36C2, D14⋊C4⋊7C22, C4.D28⋊8C2, C42⋊2C2⋊2D7, C22⋊D28⋊27C2, D14⋊D4⋊44C2, D28⋊C4⋊39C2, D14⋊Q8⋊40C2, (C2×D28)⋊29C22, C4⋊Dic7⋊61C22, C22⋊C4.40D14, D14.12(C4○D4), D14.5D4⋊38C2, D14.D4⋊48C2, (C2×C28).193C23, (C2×C14).248C24, Dic7⋊C4⋊27C22, C7⋊9(C22.32C24), (C4×Dic7)⋊38C22, C2.62(D4⋊8D14), C23.54(C22×D7), Dic7.D4⋊44C2, (C2×Dic14)⋊11C22, (C22×C14).62C23, (C23×D7).68C22, C22.269(C23×D7), C23.D7.64C22, (C2×Dic7).264C23, (C22×D7).111C23, C2.95(D7×C4○D4), (C2×C4×D7)⋊27C22, C4⋊C4⋊D7⋊41C2, (C7×C4⋊C4)⋊32C22, (D7×C22⋊C4)⋊20C2, (C7×C42⋊2C2)⋊3C2, C14.206(C2×C4○D4), (C2×C4).85(C22×D7), (C2×C7⋊D4).68C22, (C7×C22⋊C4).73C22, SmallGroup(448,1157)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1484 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C7, C2×C4 [×6], C2×C4 [×8], D4 [×9], Q8, C23, C23 [×8], D7 [×5], C14 [×3], C14, C42, C42, C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×7], C2×Q8, C24, Dic7 [×4], C28 [×6], D14 [×2], D14 [×15], C2×C14, C2×C14 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2, C42⋊2C2, Dic14, C4×D7 [×4], D28 [×7], C2×Dic7 [×4], C7⋊D4 [×2], C2×C28 [×6], C22×D7 [×4], C22×D7 [×4], C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4 [×10], C23.D7, C4×C28, C7×C22⋊C4 [×3], C7×C4⋊C4 [×3], C2×Dic14, C2×C4×D7 [×4], C2×D28 [×5], C2×C7⋊D4 [×2], C23×D7, C4×D28, C4.D28, D7×C22⋊C4, C22⋊D28 [×2], D14.D4, D14⋊D4, Dic7.D4, D28⋊C4, D14.5D4, C28⋊1D4 [×2], D14⋊Q8, C4⋊C4⋊D7, C7×C42⋊2C2, C42⋊23D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.32C24, C23×D7, D7×C4○D4, D4⋊8D14 [×2], C42⋊23D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >
►On 112 points
(1 36 12 61)(2 30 13 69)(3 38 14 63)(4 32 8 57)(5 40 9 65)(6 34 10 59)(7 42 11 67)(15 39 28 64)(16 33 22 58)(17 41 23 66)(18 35 24 60)(19 29 25 68)(20 37 26 62)(21 31 27 70)(43 111 50 84)(44 78 51 105)(45 99 52 72)(46 80 53 107)(47 101 54 74)(48 82 55 109)(49 103 56 76)(71 88 112 95)(73 90 100 97)(75 92 102 85)(77 94 104 87)(79 96 106 89)(81 98 108 91)(83 86 110 93)
(1 48 25 85)(2 56 26 93)(3 50 27 87)(4 44 28 95)(5 52 22 89)(6 46 23 97)(7 54 24 91)(8 51 15 88)(9 45 16 96)(10 53 17 90)(11 47 18 98)(12 55 19 92)(13 49 20 86)(14 43 21 94)(29 102 61 109)(30 76 62 83)(31 104 63 111)(32 78 64 71)(33 106 65 99)(34 80 66 73)(35 108 67 101)(36 82 68 75)(37 110 69 103)(38 84 70 77)(39 112 57 105)(40 72 58 79)(41 100 59 107)(42 74 60 81)
(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 7)(2 6)(3 5)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(22 27)(23 26)(24 25)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(55 98)(56 97)(57 64)(58 63)(59 62)(60 61)(65 70)(66 69)(67 68)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)
G:=sub<Sym(112)| (1,36,12,61)(2,30,13,69)(3,38,14,63)(4,32,8,57)(5,40,9,65)(6,34,10,59)(7,42,11,67)(15,39,28,64)(16,33,22,58)(17,41,23,66)(18,35,24,60)(19,29,25,68)(20,37,26,62)(21,31,27,70)(43,111,50,84)(44,78,51,105)(45,99,52,72)(46,80,53,107)(47,101,54,74)(48,82,55,109)(49,103,56,76)(71,88,112,95)(73,90,100,97)(75,92,102,85)(77,94,104,87)(79,96,106,89)(81,98,108,91)(83,86,110,93), (1,48,25,85)(2,56,26,93)(3,50,27,87)(4,44,28,95)(5,52,22,89)(6,46,23,97)(7,54,24,91)(8,51,15,88)(9,45,16,96)(10,53,17,90)(11,47,18,98)(12,55,19,92)(13,49,20,86)(14,43,21,94)(29,102,61,109)(30,76,62,83)(31,104,63,111)(32,78,64,71)(33,106,65,99)(34,80,66,73)(35,108,67,101)(36,82,68,75)(37,110,69,103)(38,84,70,77)(39,112,57,105)(40,72,58,79)(41,100,59,107)(42,74,60,81), (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), (1,7)(2,6)(3,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(22,27)(23,26)(24,25)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,98)(56,97)(57,64)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)>;
G:=Group( (1,36,12,61)(2,30,13,69)(3,38,14,63)(4,32,8,57)(5,40,9,65)(6,34,10,59)(7,42,11,67)(15,39,28,64)(16,33,22,58)(17,41,23,66)(18,35,24,60)(19,29,25,68)(20,37,26,62)(21,31,27,70)(43,111,50,84)(44,78,51,105)(45,99,52,72)(46,80,53,107)(47,101,54,74)(48,82,55,109)(49,103,56,76)(71,88,112,95)(73,90,100,97)(75,92,102,85)(77,94,104,87)(79,96,106,89)(81,98,108,91)(83,86,110,93), (1,48,25,85)(2,56,26,93)(3,50,27,87)(4,44,28,95)(5,52,22,89)(6,46,23,97)(7,54,24,91)(8,51,15,88)(9,45,16,96)(10,53,17,90)(11,47,18,98)(12,55,19,92)(13,49,20,86)(14,43,21,94)(29,102,61,109)(30,76,62,83)(31,104,63,111)(32,78,64,71)(33,106,65,99)(34,80,66,73)(35,108,67,101)(36,82,68,75)(37,110,69,103)(38,84,70,77)(39,112,57,105)(40,72,58,79)(41,100,59,107)(42,74,60,81), (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), (1,7)(2,6)(3,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(22,27)(23,26)(24,25)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,98)(56,97)(57,64)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99) );
G=PermutationGroup([(1,36,12,61),(2,30,13,69),(3,38,14,63),(4,32,8,57),(5,40,9,65),(6,34,10,59),(7,42,11,67),(15,39,28,64),(16,33,22,58),(17,41,23,66),(18,35,24,60),(19,29,25,68),(20,37,26,62),(21,31,27,70),(43,111,50,84),(44,78,51,105),(45,99,52,72),(46,80,53,107),(47,101,54,74),(48,82,55,109),(49,103,56,76),(71,88,112,95),(73,90,100,97),(75,92,102,85),(77,94,104,87),(79,96,106,89),(81,98,108,91),(83,86,110,93)], [(1,48,25,85),(2,56,26,93),(3,50,27,87),(4,44,28,95),(5,52,22,89),(6,46,23,97),(7,54,24,91),(8,51,15,88),(9,45,16,96),(10,53,17,90),(11,47,18,98),(12,55,19,92),(13,49,20,86),(14,43,21,94),(29,102,61,109),(30,76,62,83),(31,104,63,111),(32,78,64,71),(33,106,65,99),(34,80,66,73),(35,108,67,101),(36,82,68,75),(37,110,69,103),(38,84,70,77),(39,112,57,105),(40,72,58,79),(41,100,59,107),(42,74,60,81)], [(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)], [(1,7),(2,6),(3,5),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(22,27),(23,26),(24,25),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(49,90),(50,89),(51,88),(52,87),(53,86),(54,85),(55,98),(56,97),(57,64),(58,63),(59,62),(60,61),(65,70),(66,69),(67,68),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,105),(79,104),(80,103),(81,102),(82,101),(83,100),(84,99)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 0 | 28 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 8 | 8 | 8 | 8 |
| 0 | 0 | 21 | 3 | 21 | 3 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 8 | 8 | 8 | 8 |
| 0 | 0 | 3 | 21 | 3 | 21 |
G:=sub<GL(6,GF(29))| [17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,27,0,1,0,0,0,0,27,0,1],[28,2,0,0,0,0,28,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,27,18,0,0,0,0,11,2],[1,27,0,0,0,0,0,28,0,0,0,0,0,0,21,8,8,21,0,0,21,26,8,3,0,0,0,0,8,21,0,0,0,0,8,3],[28,2,0,0,0,0,0,1,0,0,0,0,0,0,21,26,8,3,0,0,21,8,8,21,0,0,0,0,8,3,0,0,0,0,8,21] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2+ (1+4) | D7×C4○D4 | D4⋊8D14 |
| kernel | C42⋊23D14 | C4×D28 | C4.D28 | D7×C22⋊C4 | C22⋊D28 | D14.D4 | D14⋊D4 | Dic7.D4 | D28⋊C4 | D14.5D4 | C28⋊1D4 | D14⋊Q8 | C4⋊C4⋊D7 | C7×C42⋊2C2 | C42⋊2C2 | D14 | C42 | C22⋊C4 | C4⋊C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 4 | 3 | 9 | 9 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{23}D_{14} % in TeX
G:=Group("C4^2:23D14"); // GroupNames label
G:=SmallGroup(448,1157);
// by ID
G=gap.SmallGroup(448,1157);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations