metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊21D14, (C4×D7)⋊12D4, (C2×D4)⋊22D14, C4⋊D4⋊26D7, C4.182(D4×D7), D14⋊3(C4○D4), C23⋊D14⋊8C2, C28⋊2D4⋊17C2, C22⋊C4⋊26D14, (D4×Dic7)⋊20C2, D14.40(C2×D4), C28.226(C2×D4), D14⋊2Q8⋊20C2, Dic7⋊4D4⋊8C2, (D4×C14)⋊11C22, C4⋊Dic7⋊30C22, Dic7.63(C2×D4), C14.65(C22×D4), D14.D4⋊18C2, C28.48D4⋊33C2, C22⋊2(D4⋊2D7), (C2×C28).594C23, (C2×C14).150C24, Dic7⋊C4⋊28C22, D14⋊C4.14C22, C7⋊4(C22.19C24), (C4×Dic7)⋊20C22, (C22×C4).368D14, C23.D7⋊22C22, C23.15(C22×D7), (C2×Dic14)⋊24C22, (C22×C14).19C23, (C2×Dic7).71C23, C22.171(C23×D7), (C22×C28).240C22, (C22×Dic7)⋊19C22, (C23×D7).107C22, (C22×D7).185C23, C2.38(C2×D4×D7), (D7×C22×C4)⋊4C2, (C7×C4⋊C4)⋊9C22, C2.38(D7×C4○D4), C4⋊C4⋊7D7⋊19C2, (C2×C14)⋊5(C4○D4), (C7×C4⋊D4)⋊12C2, (C2×D4⋊2D7)⋊12C2, C14.151(C2×C4○D4), C2.36(C2×D4⋊2D7), (C2×C4×D7).247C22, (C7×C22⋊C4)⋊11C22, (C2×C4).294(C22×D7), (C2×C7⋊D4).27C22, SmallGroup(448,1059)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1548 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×24], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D7 [×4], C14 [×3], C14 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×11], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic7 [×2], Dic7 [×5], C28 [×2], C28 [×3], D14 [×4], D14 [×12], C2×C14, C2×C14 [×2], C2×C14 [×8], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic14 [×2], C4×D7 [×4], C4×D7 [×6], C2×Dic7 [×2], C2×Dic7 [×4], C2×Dic7 [×6], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×6], C22×D7 [×2], C22×D7 [×6], C22×C14, C22×C14 [×2], C22.19C24, C4×Dic7 [×2], Dic7⋊C4 [×2], C4⋊Dic7, C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7 [×4], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7 [×4], C2×C4×D7 [×4], D4⋊2D7 [×4], C22×Dic7, C22×Dic7 [×2], C2×C7⋊D4 [×4], C22×C28, D4×C14, D4×C14 [×2], C23×D7, Dic7⋊4D4 [×2], D14.D4 [×2], C4⋊C4⋊7D7, D14⋊2Q8, C28.48D4, D4×Dic7 [×2], C23⋊D14 [×2], C28⋊2D4, C7×C4⋊D4, D7×C22×C4, C2×D4⋊2D7, C4⋊C4⋊21D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C22×D7 [×7], C22.19C24, D4×D7 [×2], D4⋊2D7 [×2], C23×D7, C2×D4×D7, C2×D4⋊2D7, D7×C4○D4, C4⋊C4⋊21D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd=a2b, dcd=c-1 >
►On 112 points
(1 14 17 30)(2 8 18 31)(3 9 19 32)(4 10 20 33)(5 11 21 34)(6 12 15 35)(7 13 16 29)(22 36 44 54)(23 37 45 55)(24 38 46 56)(25 39 47 50)(26 40 48 51)(27 41 49 52)(28 42 43 53)(57 103 94 75)(58 104 95 76)(59 105 96 77)(60 106 97 78)(61 107 98 79)(62 108 85 80)(63 109 86 81)(64 110 87 82)(65 111 88 83)(66 112 89 84)(67 99 90 71)(68 100 91 72)(69 101 92 73)(70 102 93 74)
(1 108 22 101)(2 102 23 109)(3 110 24 103)(4 104 25 111)(5 112 26 105)(6 106 27 99)(7 100 28 107)(8 70 37 63)(9 64 38 57)(10 58 39 65)(11 66 40 59)(12 60 41 67)(13 68 42 61)(14 62 36 69)(15 78 49 71)(16 72 43 79)(17 80 44 73)(18 74 45 81)(19 82 46 75)(20 76 47 83)(21 84 48 77)(29 91 53 98)(30 85 54 92)(31 93 55 86)(32 87 56 94)(33 95 50 88)(34 89 51 96)(35 97 52 90)
(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)(8 12)(9 11)(13 14)(15 18)(16 17)(19 21)(22 28)(23 27)(24 26)(29 30)(31 35)(32 34)(36 42)(37 41)(38 40)(43 44)(45 49)(46 48)(51 56)(52 55)(53 54)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 98)(70 97)(71 109)(72 108)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 112)(83 111)(84 110)
G:=sub<Sym(112)| (1,14,17,30)(2,8,18,31)(3,9,19,32)(4,10,20,33)(5,11,21,34)(6,12,15,35)(7,13,16,29)(22,36,44,54)(23,37,45,55)(24,38,46,56)(25,39,47,50)(26,40,48,51)(27,41,49,52)(28,42,43,53)(57,103,94,75)(58,104,95,76)(59,105,96,77)(60,106,97,78)(61,107,98,79)(62,108,85,80)(63,109,86,81)(64,110,87,82)(65,111,88,83)(66,112,89,84)(67,99,90,71)(68,100,91,72)(69,101,92,73)(70,102,93,74), (1,108,22,101)(2,102,23,109)(3,110,24,103)(4,104,25,111)(5,112,26,105)(6,106,27,99)(7,100,28,107)(8,70,37,63)(9,64,38,57)(10,58,39,65)(11,66,40,59)(12,60,41,67)(13,68,42,61)(14,62,36,69)(15,78,49,71)(16,72,43,79)(17,80,44,73)(18,74,45,81)(19,82,46,75)(20,76,47,83)(21,84,48,77)(29,91,53,98)(30,85,54,92)(31,93,55,86)(32,87,56,94)(33,95,50,88)(34,89,51,96)(35,97,52,90), (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)(8,12)(9,11)(13,14)(15,18)(16,17)(19,21)(22,28)(23,27)(24,26)(29,30)(31,35)(32,34)(36,42)(37,41)(38,40)(43,44)(45,49)(46,48)(51,56)(52,55)(53,54)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,98)(70,97)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,112)(83,111)(84,110)>;
G:=Group( (1,14,17,30)(2,8,18,31)(3,9,19,32)(4,10,20,33)(5,11,21,34)(6,12,15,35)(7,13,16,29)(22,36,44,54)(23,37,45,55)(24,38,46,56)(25,39,47,50)(26,40,48,51)(27,41,49,52)(28,42,43,53)(57,103,94,75)(58,104,95,76)(59,105,96,77)(60,106,97,78)(61,107,98,79)(62,108,85,80)(63,109,86,81)(64,110,87,82)(65,111,88,83)(66,112,89,84)(67,99,90,71)(68,100,91,72)(69,101,92,73)(70,102,93,74), (1,108,22,101)(2,102,23,109)(3,110,24,103)(4,104,25,111)(5,112,26,105)(6,106,27,99)(7,100,28,107)(8,70,37,63)(9,64,38,57)(10,58,39,65)(11,66,40,59)(12,60,41,67)(13,68,42,61)(14,62,36,69)(15,78,49,71)(16,72,43,79)(17,80,44,73)(18,74,45,81)(19,82,46,75)(20,76,47,83)(21,84,48,77)(29,91,53,98)(30,85,54,92)(31,93,55,86)(32,87,56,94)(33,95,50,88)(34,89,51,96)(35,97,52,90), (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)(8,12)(9,11)(13,14)(15,18)(16,17)(19,21)(22,28)(23,27)(24,26)(29,30)(31,35)(32,34)(36,42)(37,41)(38,40)(43,44)(45,49)(46,48)(51,56)(52,55)(53,54)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,98)(70,97)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,112)(83,111)(84,110) );
G=PermutationGroup([(1,14,17,30),(2,8,18,31),(3,9,19,32),(4,10,20,33),(5,11,21,34),(6,12,15,35),(7,13,16,29),(22,36,44,54),(23,37,45,55),(24,38,46,56),(25,39,47,50),(26,40,48,51),(27,41,49,52),(28,42,43,53),(57,103,94,75),(58,104,95,76),(59,105,96,77),(60,106,97,78),(61,107,98,79),(62,108,85,80),(63,109,86,81),(64,110,87,82),(65,111,88,83),(66,112,89,84),(67,99,90,71),(68,100,91,72),(69,101,92,73),(70,102,93,74)], [(1,108,22,101),(2,102,23,109),(3,110,24,103),(4,104,25,111),(5,112,26,105),(6,106,27,99),(7,100,28,107),(8,70,37,63),(9,64,38,57),(10,58,39,65),(11,66,40,59),(12,60,41,67),(13,68,42,61),(14,62,36,69),(15,78,49,71),(16,72,43,79),(17,80,44,73),(18,74,45,81),(19,82,46,75),(20,76,47,83),(21,84,48,77),(29,91,53,98),(30,85,54,92),(31,93,55,86),(32,87,56,94),(33,95,50,88),(34,89,51,96),(35,97,52,90)], [(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),(8,12),(9,11),(13,14),(15,18),(16,17),(19,21),(22,28),(23,27),(24,26),(29,30),(31,35),(32,34),(36,42),(37,41),(38,40),(43,44),(45,49),(46,48),(51,56),(52,55),(53,54),(57,96),(58,95),(59,94),(60,93),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,98),(70,97),(71,109),(72,108),(73,107),(74,106),(75,105),(76,104),(77,103),(78,102),(79,101),(80,100),(81,99),(82,112),(83,111),(84,110)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 17 | 2 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 21 | 0 | 0 |
| 0 | 0 | 16 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 28 | 0 | 0 |
| 0 | 0 | 5 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [17,1,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,12],[17,1,0,0,0,0,2,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,16,0,0,0,0,21,20,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,12,0,0,0,0,0,28,0,0,0,0,0,0,8,5,0,0,0,0,28,21,0,0,0,0,0,0,1,0,0,0,0,0,0,28] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D4×D7 | D4⋊2D7 | D7×C4○D4 |
| kernel | C4⋊C4⋊21D14 | Dic7⋊4D4 | D14.D4 | C4⋊C4⋊7D7 | D14⋊2Q8 | C28.48D4 | D4×Dic7 | C23⋊D14 | C28⋊2D4 | C7×C4⋊D4 | D7×C22×C4 | C2×D4⋊2D7 | C4×D7 | C4⋊D4 | D14 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 4 | 6 | 3 | 3 | 9 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{21}D_{14} % in TeX
G:=Group("C4:C4:21D14"); // GroupNames label
G:=SmallGroup(448,1059);
// by ID
G=gap.SmallGroup(448,1059);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,1123,794,297,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations