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, (C22×D7).185C23, (C23×D7).107C22, 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
Generators and relations for C4⋊C4⋊21D14
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 >
Subgroups: 1548 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22.19C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, D4⋊2D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, 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⋊C4⋊21D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, C22×D7, C22.19C24, D4×D7, D4⋊2D7, C23×D7, C2×D4×D7, C2×D4⋊2D7, D7×C4○D4, C4⋊C4⋊21D14
►On 112 points
(1 47 19 42)(2 48 20 36)(3 49 21 37)(4 43 15 38)(5 44 16 39)(6 45 17 40)(7 46 18 41)(8 52 22 35)(9 53 23 29)(10 54 24 30)(11 55 25 31)(12 56 26 32)(13 50 27 33)(14 51 28 34)(57 77 106 91)(58 78 107 92)(59 79 108 93)(60 80 109 94)(61 81 110 95)(62 82 111 96)(63 83 112 97)(64 84 99 98)(65 71 100 85)(66 72 101 86)(67 73 102 87)(68 74 103 88)(69 75 104 89)(70 76 105 90)
(1 62 26 69)(2 70 27 63)(3 64 28 57)(4 58 22 65)(5 66 23 59)(6 60 24 67)(7 68 25 61)(8 100 15 107)(9 108 16 101)(10 102 17 109)(11 110 18 103)(12 104 19 111)(13 112 20 105)(14 106 21 99)(29 93 44 86)(30 87 45 94)(31 95 46 88)(32 89 47 96)(33 97 48 90)(34 91 49 98)(35 85 43 92)(36 76 50 83)(37 84 51 77)(38 78 52 71)(39 72 53 79)(40 80 54 73)(41 74 55 81)(42 82 56 75)
(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)(23 28)(24 27)(25 26)(29 34)(30 33)(31 32)(36 40)(37 39)(41 42)(44 49)(45 48)(46 47)(50 54)(51 53)(55 56)(57 108)(58 107)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 112)(68 111)(69 110)(70 109)(71 85)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)
G:=sub<Sym(112)| (1,47,19,42)(2,48,20,36)(3,49,21,37)(4,43,15,38)(5,44,16,39)(6,45,17,40)(7,46,18,41)(8,52,22,35)(9,53,23,29)(10,54,24,30)(11,55,25,31)(12,56,26,32)(13,50,27,33)(14,51,28,34)(57,77,106,91)(58,78,107,92)(59,79,108,93)(60,80,109,94)(61,81,110,95)(62,82,111,96)(63,83,112,97)(64,84,99,98)(65,71,100,85)(66,72,101,86)(67,73,102,87)(68,74,103,88)(69,75,104,89)(70,76,105,90), (1,62,26,69)(2,70,27,63)(3,64,28,57)(4,58,22,65)(5,66,23,59)(6,60,24,67)(7,68,25,61)(8,100,15,107)(9,108,16,101)(10,102,17,109)(11,110,18,103)(12,104,19,111)(13,112,20,105)(14,106,21,99)(29,93,44,86)(30,87,45,94)(31,95,46,88)(32,89,47,96)(33,97,48,90)(34,91,49,98)(35,85,43,92)(36,76,50,83)(37,84,51,77)(38,78,52,71)(39,72,53,79)(40,80,54,73)(41,74,55,81)(42,82,56,75), (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)(23,28)(24,27)(25,26)(29,34)(30,33)(31,32)(36,40)(37,39)(41,42)(44,49)(45,48)(46,47)(50,54)(51,53)(55,56)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,112)(68,111)(69,110)(70,109)(71,85)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)>;
G:=Group( (1,47,19,42)(2,48,20,36)(3,49,21,37)(4,43,15,38)(5,44,16,39)(6,45,17,40)(7,46,18,41)(8,52,22,35)(9,53,23,29)(10,54,24,30)(11,55,25,31)(12,56,26,32)(13,50,27,33)(14,51,28,34)(57,77,106,91)(58,78,107,92)(59,79,108,93)(60,80,109,94)(61,81,110,95)(62,82,111,96)(63,83,112,97)(64,84,99,98)(65,71,100,85)(66,72,101,86)(67,73,102,87)(68,74,103,88)(69,75,104,89)(70,76,105,90), (1,62,26,69)(2,70,27,63)(3,64,28,57)(4,58,22,65)(5,66,23,59)(6,60,24,67)(7,68,25,61)(8,100,15,107)(9,108,16,101)(10,102,17,109)(11,110,18,103)(12,104,19,111)(13,112,20,105)(14,106,21,99)(29,93,44,86)(30,87,45,94)(31,95,46,88)(32,89,47,96)(33,97,48,90)(34,91,49,98)(35,85,43,92)(36,76,50,83)(37,84,51,77)(38,78,52,71)(39,72,53,79)(40,80,54,73)(41,74,55,81)(42,82,56,75), (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)(23,28)(24,27)(25,26)(29,34)(30,33)(31,32)(36,40)(37,39)(41,42)(44,49)(45,48)(46,47)(50,54)(51,53)(55,56)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,112)(68,111)(69,110)(70,109)(71,85)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86) );
G=PermutationGroup([[(1,47,19,42),(2,48,20,36),(3,49,21,37),(4,43,15,38),(5,44,16,39),(6,45,17,40),(7,46,18,41),(8,52,22,35),(9,53,23,29),(10,54,24,30),(11,55,25,31),(12,56,26,32),(13,50,27,33),(14,51,28,34),(57,77,106,91),(58,78,107,92),(59,79,108,93),(60,80,109,94),(61,81,110,95),(62,82,111,96),(63,83,112,97),(64,84,99,98),(65,71,100,85),(66,72,101,86),(67,73,102,87),(68,74,103,88),(69,75,104,89),(70,76,105,90)], [(1,62,26,69),(2,70,27,63),(3,64,28,57),(4,58,22,65),(5,66,23,59),(6,60,24,67),(7,68,25,61),(8,100,15,107),(9,108,16,101),(10,102,17,109),(11,110,18,103),(12,104,19,111),(13,112,20,105),(14,106,21,99),(29,93,44,86),(30,87,45,94),(31,95,46,88),(32,89,47,96),(33,97,48,90),(34,91,49,98),(35,85,43,92),(36,76,50,83),(37,84,51,77),(38,78,52,71),(39,72,53,79),(40,80,54,73),(41,74,55,81),(42,82,56,75)], [(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),(23,28),(24,27),(25,26),(29,34),(30,33),(31,32),(36,40),(37,39),(41,42),(44,49),(45,48),(46,47),(50,54),(51,53),(55,56),(57,108),(58,107),(59,106),(60,105),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,112),(68,111),(69,110),(70,109),(71,85),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86)]])
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 |
Matrix representation of C4⋊C4⋊21D14 ►in 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] >;
C4⋊C4⋊21D14 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