metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊21D14, C14.732+ (1+4), (C2×Q8)⋊9D14, (C4×C28)⋊33C22, C22⋊C4⋊35D14, C4.4D4⋊13D7, D14⋊3Q8⋊31C2, C23⋊D14.6C2, (C2×D4).111D14, C42⋊D7⋊36C2, C4⋊Dic7⋊42C22, (Q8×C14)⋊15C22, Dic7⋊4D4⋊32C2, D14.23(C4○D4), D14.D4⋊44C2, (C2×C14).223C24, (C2×C28).632C23, Dic7⋊C4⋊67C22, (C4×Dic7)⋊57C22, C2.76(D4⋊6D14), C23.D7⋊34C22, C23.45(C22×D7), D14⋊C4.136C22, C7⋊8(C22.45C24), (D4×C14).211C22, C23.D14⋊40C2, (C22×C14).53C23, (C23×D7).66C22, C22.244(C23×D7), C23.18D14⋊25C2, (C2×Dic7).255C23, (C22×Dic7)⋊28C22, (C22×D7).217C23, C2.79(D7×C4○D4), (D7×C22⋊C4)⋊19C2, C14.190(C2×C4○D4), (C7×C4.4D4)⋊15C2, (C2×C4×D7).215C22, (C2×C4).74(C22×D7), (C7×C22⋊C4)⋊31C22, (C2×C7⋊D4).61C22, SmallGroup(448,1132)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1228 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×18], C7, C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D7 [×4], C14, C14 [×2], C14 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic7 [×6], C28 [×5], D14 [×4], D14 [×8], C2×C14, C2×C14 [×6], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], C4×D7 [×6], C2×Dic7 [×6], C2×Dic7, C7⋊D4 [×4], C2×C28, C2×C28 [×4], C7×D4, C7×Q8, C22×D7 [×2], C22×D7 [×5], C22×C14 [×2], C22.45C24, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×2], D14⋊C4 [×6], C23.D7 [×2], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×4], C2×C4×D7 [×4], C22×Dic7, C2×C7⋊D4 [×2], D4×C14, Q8×C14, C23×D7, C42⋊D7 [×2], C23.D14 [×2], D7×C22⋊C4 [×2], Dic7⋊4D4 [×2], D14.D4 [×2], C23.18D14, C23⋊D14, D14⋊3Q8 [×2], C7×C4.4D4, C42⋊21D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.45C24, C23×D7, D4⋊6D14, D7×C4○D4 [×2], C42⋊21D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 112 points
(1 83 19 63)(2 77 20 57)(3 71 21 65)(4 79 15 59)(5 73 16 67)(6 81 17 61)(7 75 18 69)(8 76 26 70)(9 84 27 64)(10 78 28 58)(11 72 22 66)(12 80 23 60)(13 74 24 68)(14 82 25 62)(29 87 36 109)(30 103 37 95)(31 89 38 111)(32 105 39 97)(33 91 40 99)(34 107 41 85)(35 93 42 101)(43 108 50 86)(44 94 51 102)(45 110 52 88)(46 96 53 104)(47 112 54 90)(48 98 55 106)(49 100 56 92)
(1 34 8 56)(2 42 9 50)(3 36 10 44)(4 30 11 52)(5 38 12 46)(6 32 13 54)(7 40 14 48)(15 37 22 45)(16 31 23 53)(17 39 24 47)(18 33 25 55)(19 41 26 49)(20 35 27 43)(21 29 28 51)(57 93 64 108)(58 102 65 87)(59 95 66 110)(60 104 67 89)(61 97 68 112)(62 106 69 91)(63 85 70 100)(71 109 78 94)(72 88 79 103)(73 111 80 96)(74 90 81 105)(75 99 82 98)(76 92 83 107)(77 101 84 86)
(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 14)(9 13)(10 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 38)(30 37)(31 36)(32 35)(33 34)(39 42)(40 41)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(55 56)(57 61)(58 60)(62 70)(63 69)(64 68)(65 67)(71 73)(74 84)(75 83)(76 82)(77 81)(78 80)(85 99)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
G:=sub<Sym(112)| (1,83,19,63)(2,77,20,57)(3,71,21,65)(4,79,15,59)(5,73,16,67)(6,81,17,61)(7,75,18,69)(8,76,26,70)(9,84,27,64)(10,78,28,58)(11,72,22,66)(12,80,23,60)(13,74,24,68)(14,82,25,62)(29,87,36,109)(30,103,37,95)(31,89,38,111)(32,105,39,97)(33,91,40,99)(34,107,41,85)(35,93,42,101)(43,108,50,86)(44,94,51,102)(45,110,52,88)(46,96,53,104)(47,112,54,90)(48,98,55,106)(49,100,56,92), (1,34,8,56)(2,42,9,50)(3,36,10,44)(4,30,11,52)(5,38,12,46)(6,32,13,54)(7,40,14,48)(15,37,22,45)(16,31,23,53)(17,39,24,47)(18,33,25,55)(19,41,26,49)(20,35,27,43)(21,29,28,51)(57,93,64,108)(58,102,65,87)(59,95,66,110)(60,104,67,89)(61,97,68,112)(62,106,69,91)(63,85,70,100)(71,109,78,94)(72,88,79,103)(73,111,80,96)(74,90,81,105)(75,99,82,98)(76,92,83,107)(77,101,84,86), (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)(8,14)(9,13)(10,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,38)(30,37)(31,36)(32,35)(33,34)(39,42)(40,41)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(55,56)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,73)(74,84)(75,83)(76,82)(77,81)(78,80)(85,99)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)>;
G:=Group( (1,83,19,63)(2,77,20,57)(3,71,21,65)(4,79,15,59)(5,73,16,67)(6,81,17,61)(7,75,18,69)(8,76,26,70)(9,84,27,64)(10,78,28,58)(11,72,22,66)(12,80,23,60)(13,74,24,68)(14,82,25,62)(29,87,36,109)(30,103,37,95)(31,89,38,111)(32,105,39,97)(33,91,40,99)(34,107,41,85)(35,93,42,101)(43,108,50,86)(44,94,51,102)(45,110,52,88)(46,96,53,104)(47,112,54,90)(48,98,55,106)(49,100,56,92), (1,34,8,56)(2,42,9,50)(3,36,10,44)(4,30,11,52)(5,38,12,46)(6,32,13,54)(7,40,14,48)(15,37,22,45)(16,31,23,53)(17,39,24,47)(18,33,25,55)(19,41,26,49)(20,35,27,43)(21,29,28,51)(57,93,64,108)(58,102,65,87)(59,95,66,110)(60,104,67,89)(61,97,68,112)(62,106,69,91)(63,85,70,100)(71,109,78,94)(72,88,79,103)(73,111,80,96)(74,90,81,105)(75,99,82,98)(76,92,83,107)(77,101,84,86), (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)(8,14)(9,13)(10,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,38)(30,37)(31,36)(32,35)(33,34)(39,42)(40,41)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(55,56)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,73)(74,84)(75,83)(76,82)(77,81)(78,80)(85,99)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100) );
G=PermutationGroup([(1,83,19,63),(2,77,20,57),(3,71,21,65),(4,79,15,59),(5,73,16,67),(6,81,17,61),(7,75,18,69),(8,76,26,70),(9,84,27,64),(10,78,28,58),(11,72,22,66),(12,80,23,60),(13,74,24,68),(14,82,25,62),(29,87,36,109),(30,103,37,95),(31,89,38,111),(32,105,39,97),(33,91,40,99),(34,107,41,85),(35,93,42,101),(43,108,50,86),(44,94,51,102),(45,110,52,88),(46,96,53,104),(47,112,54,90),(48,98,55,106),(49,100,56,92)], [(1,34,8,56),(2,42,9,50),(3,36,10,44),(4,30,11,52),(5,38,12,46),(6,32,13,54),(7,40,14,48),(15,37,22,45),(16,31,23,53),(17,39,24,47),(18,33,25,55),(19,41,26,49),(20,35,27,43),(21,29,28,51),(57,93,64,108),(58,102,65,87),(59,95,66,110),(60,104,67,89),(61,97,68,112),(62,106,69,91),(63,85,70,100),(71,109,78,94),(72,88,79,103),(73,111,80,96),(74,90,81,105),(75,99,82,98),(76,92,83,107),(77,101,84,86)], [(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),(8,14),(9,13),(10,12),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,38),(30,37),(31,36),(32,35),(33,34),(39,42),(40,41),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(55,56),(57,61),(58,60),(62,70),(63,69),(64,68),(65,67),(71,73),(74,84),(75,83),(76,82),(77,81),(78,80),(85,99),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 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 | 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 28 | 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 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 8 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 26 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,27,1],[1,0,0,0,0,0,28,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,10,12,0,0,0,0,8,1,0,0,0,0,0,0,1,28,0,0,0,0,0,28],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,22,16,0,0,0,0,26,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 |
| kernel | C42⋊21D14 | C42⋊D7 | C23.D14 | D7×C22⋊C4 | Dic7⋊4D4 | D14.D4 | C23.18D14 | C23⋊D14 | D14⋊3Q8 | C7×C4.4D4 | C4.4D4 | D14 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 8 | 3 | 12 | 3 | 3 | 1 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{21}D_{14} % in TeX
G:=Group("C4^2:21D14"); // GroupNames label
G:=SmallGroup(448,1132);
// by ID
G=gap.SmallGroup(448,1132);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,387,100,346,136,18822]);
// 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*b^2,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations