direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×D4⋊5D4, C14.1612+ 1+4, D4⋊5(C7×D4), (C7×D4)⋊23D4, (C4×D4)⋊15C14, (D4×C28)⋊44C2, C42⋊9(C2×C14), C4.40(D4×C14), C22≀C2⋊6C14, C4⋊D4⋊11C14, (C4×C28)⋊43C22, C28.401(C2×D4), (C22×D4)⋊9C14, C22⋊Q8⋊11C14, C22.5(D4×C14), C4.4D4⋊10C14, (D4×C14)⋊38C22, C24.22(C2×C14), (Q8×C14)⋊52C22, (C2×C14).366C24, (C2×C28).713C23, (C22×C28)⋊51C22, C14.194(C22×D4), C22.D4⋊8C14, (C22×C14).98C23, C22.40(C23×C14), (C23×C14).19C22, C23.16(C22×C14), C2.13(C7×2+ 1+4), C4⋊C4⋊5(C2×C14), (D4×C2×C14)⋊24C2, C2.18(D4×C2×C14), (C2×C4○D4)⋊6C14, C22⋊2(C7×C4○D4), (C14×C4○D4)⋊22C2, (C2×D4)⋊13(C2×C14), (C7×C4⋊D4)⋊38C2, (C7×C4⋊C4)⋊73C22, (C2×Q8)⋊12(C2×C14), C2.20(C14×C4○D4), (C7×C22⋊Q8)⋊38C2, (C7×C22≀C2)⋊16C2, (C2×C14)⋊14(C4○D4), (C14×C22⋊C4)⋊34C2, (C2×C22⋊C4)⋊14C14, C22⋊C4⋊16(C2×C14), (C22×C4)⋊11(C2×C14), C14.239(C2×C4○D4), (C2×C14).182(C2×D4), (C7×C4.4D4)⋊30C2, (C7×C22⋊C4)⋊70C22, (C2×C4).59(C22×C14), (C7×C22.D4)⋊27C2, SmallGroup(448,1329)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4⋊5D4
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 570 in 334 conjugacy classes, 166 normal (62 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, 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, C28, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, C22×C14, C22×C14, D4⋊5D4, C4×C28, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C23×C14, C14×C22⋊C4, D4×C28, C7×C22≀C2, C7×C4⋊D4, C7×C4⋊D4, C7×C22⋊Q8, C7×C22.D4, C7×C4.4D4, D4×C2×C14, C14×C4○D4, C7×D4⋊5D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C24, C2×C14, C22×D4, C2×C4○D4, 2+ 1+4, C7×D4, C22×C14, D4⋊5D4, D4×C14, C7×C4○D4, C23×C14, D4×C2×C14, C14×C4○D4, C7×2+ 1+4, C7×D4⋊5D4
►On 112 points
Generators in S112
(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 67 38 59)(2 68 39 60)(3 69 40 61)(4 70 41 62)(5 64 42 63)(6 65 36 57)(7 66 37 58)(8 105 27 81)(9 99 28 82)(10 100 22 83)(11 101 23 84)(12 102 24 78)(13 103 25 79)(14 104 26 80)(15 98 109 90)(16 92 110 91)(17 93 111 85)(18 94 112 86)(19 95 106 87)(20 96 107 88)(21 97 108 89)(29 55 44 72)(30 56 45 73)(31 50 46 74)(32 51 47 75)(33 52 48 76)(34 53 49 77)(35 54 43 71)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
(1 102 46 95)(2 103 47 96)(3 104 48 97)(4 105 49 98)(5 99 43 92)(6 100 44 93)(7 101 45 94)(8 53 15 62)(9 54 16 63)(10 55 17 57)(11 56 18 58)(12 50 19 59)(13 51 20 60)(14 52 21 61)(22 72 111 65)(23 73 112 66)(24 74 106 67)(25 75 107 68)(26 76 108 69)(27 77 109 70)(28 71 110 64)(29 85 36 83)(30 86 37 84)(31 87 38 78)(32 88 39 79)(33 89 40 80)(34 90 41 81)(35 91 42 82)
(1 95)(2 96)(3 97)(4 98)(5 92)(6 93)(7 94)(8 53)(9 54)(10 55)(11 56)(12 50)(13 51)(14 52)(15 62)(16 63)(17 57)(18 58)(19 59)(20 60)(21 61)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 71)(29 83)(30 84)(31 78)(32 79)(33 80)(34 81)(35 82)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(64 110)(65 111)(66 112)(67 106)(68 107)(69 108)(70 109)
G:=sub<Sym(112)| (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), (1,67,38,59)(2,68,39,60)(3,69,40,61)(4,70,41,62)(5,64,42,63)(6,65,36,57)(7,66,37,58)(8,105,27,81)(9,99,28,82)(10,100,22,83)(11,101,23,84)(12,102,24,78)(13,103,25,79)(14,104,26,80)(15,98,109,90)(16,92,110,91)(17,93,111,85)(18,94,112,86)(19,95,106,87)(20,96,107,88)(21,97,108,89)(29,55,44,72)(30,56,45,73)(31,50,46,74)(32,51,47,75)(33,52,48,76)(34,53,49,77)(35,54,43,71), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (1,102,46,95)(2,103,47,96)(3,104,48,97)(4,105,49,98)(5,99,43,92)(6,100,44,93)(7,101,45,94)(8,53,15,62)(9,54,16,63)(10,55,17,57)(11,56,18,58)(12,50,19,59)(13,51,20,60)(14,52,21,61)(22,72,111,65)(23,73,112,66)(24,74,106,67)(25,75,107,68)(26,76,108,69)(27,77,109,70)(28,71,110,64)(29,85,36,83)(30,86,37,84)(31,87,38,78)(32,88,39,79)(33,89,40,80)(34,90,41,81)(35,91,42,82), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,53)(9,54)(10,55)(11,56)(12,50)(13,51)(14,52)(15,62)(16,63)(17,57)(18,58)(19,59)(20,60)(21,61)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,71)(29,83)(30,84)(31,78)(32,79)(33,80)(34,81)(35,82)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(64,110)(65,111)(66,112)(67,106)(68,107)(69,108)(70,109)>;
G:=Group( (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), (1,67,38,59)(2,68,39,60)(3,69,40,61)(4,70,41,62)(5,64,42,63)(6,65,36,57)(7,66,37,58)(8,105,27,81)(9,99,28,82)(10,100,22,83)(11,101,23,84)(12,102,24,78)(13,103,25,79)(14,104,26,80)(15,98,109,90)(16,92,110,91)(17,93,111,85)(18,94,112,86)(19,95,106,87)(20,96,107,88)(21,97,108,89)(29,55,44,72)(30,56,45,73)(31,50,46,74)(32,51,47,75)(33,52,48,76)(34,53,49,77)(35,54,43,71), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (1,102,46,95)(2,103,47,96)(3,104,48,97)(4,105,49,98)(5,99,43,92)(6,100,44,93)(7,101,45,94)(8,53,15,62)(9,54,16,63)(10,55,17,57)(11,56,18,58)(12,50,19,59)(13,51,20,60)(14,52,21,61)(22,72,111,65)(23,73,112,66)(24,74,106,67)(25,75,107,68)(26,76,108,69)(27,77,109,70)(28,71,110,64)(29,85,36,83)(30,86,37,84)(31,87,38,78)(32,88,39,79)(33,89,40,80)(34,90,41,81)(35,91,42,82), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,53)(9,54)(10,55)(11,56)(12,50)(13,51)(14,52)(15,62)(16,63)(17,57)(18,58)(19,59)(20,60)(21,61)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,71)(29,83)(30,84)(31,78)(32,79)(33,80)(34,81)(35,82)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(64,110)(65,111)(66,112)(67,106)(68,107)(69,108)(70,109) );
G=PermutationGroup([[(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)], [(1,67,38,59),(2,68,39,60),(3,69,40,61),(4,70,41,62),(5,64,42,63),(6,65,36,57),(7,66,37,58),(8,105,27,81),(9,99,28,82),(10,100,22,83),(11,101,23,84),(12,102,24,78),(13,103,25,79),(14,104,26,80),(15,98,109,90),(16,92,110,91),(17,93,111,85),(18,94,112,86),(19,95,106,87),(20,96,107,88),(21,97,108,89),(29,55,44,72),(30,56,45,73),(31,50,46,74),(32,51,47,75),(33,52,48,76),(34,53,49,77),(35,54,43,71)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,67),(51,68),(52,69),(53,70),(54,64),(55,65),(56,66),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,95),(79,96),(80,97),(81,98),(82,92),(83,93),(84,94),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)], [(1,102,46,95),(2,103,47,96),(3,104,48,97),(4,105,49,98),(5,99,43,92),(6,100,44,93),(7,101,45,94),(8,53,15,62),(9,54,16,63),(10,55,17,57),(11,56,18,58),(12,50,19,59),(13,51,20,60),(14,52,21,61),(22,72,111,65),(23,73,112,66),(24,74,106,67),(25,75,107,68),(26,76,108,69),(27,77,109,70),(28,71,110,64),(29,85,36,83),(30,86,37,84),(31,87,38,78),(32,88,39,79),(33,89,40,80),(34,90,41,81),(35,91,42,82)], [(1,95),(2,96),(3,97),(4,98),(5,92),(6,93),(7,94),(8,53),(9,54),(10,55),(11,56),(12,50),(13,51),(14,52),(15,62),(16,63),(17,57),(18,58),(19,59),(20,60),(21,61),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,71),(29,83),(30,84),(31,78),(32,79),(33,80),(34,81),(35,82),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(64,110),(65,111),(66,112),(67,106),(68,107),(69,108),(70,109)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 4A | ··· | 4F | 4G | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 14BC | ··· | 14BT | 28A | ··· | 28AJ | 28AK | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | 2+ 1+4 | C7×2+ 1+4 |
| kernel | C7×D4⋊5D4 | C14×C22⋊C4 | D4×C28 | C7×C22≀C2 | C7×C4⋊D4 | C7×C22⋊Q8 | C7×C22.D4 | C7×C4.4D4 | D4×C2×C14 | C14×C4○D4 | D4⋊5D4 | C2×C22⋊C4 | C4×D4 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C22×D4 | C2×C4○D4 | C7×D4 | C2×C14 | D4 | C22 | C14 | C2 |
| # reps | 1 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 1 | 1 | 6 | 12 | 12 | 12 | 18 | 6 | 12 | 6 | 6 | 6 | 4 | 4 | 24 | 24 | 1 | 6 |
Matrix representation of C7×D4⋊5D4 ►in GL4(𝔽29) generated by
| 24 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 24 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 27 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 28 |
| 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 17 |
| 0 | 0 | 24 | 12 |
| 0 | 28 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 17 | 17 |
| 0 | 0 | 24 | 12 |
G:=sub<GL(4,GF(29))| [24,0,0,0,0,24,0,0,0,0,24,0,0,0,0,24],[28,0,0,0,0,28,0,0,0,0,1,27,0,0,1,28],[28,0,0,0,0,28,0,0,0,0,1,27,0,0,0,28],[0,1,0,0,28,0,0,0,0,0,17,24,0,0,17,12],[0,28,0,0,28,0,0,0,0,0,17,24,0,0,17,12] >;
C7×D4⋊5D4 in GAP, Magma, Sage, TeX
C_7\times D_4\rtimes_5D_4
% in TeX
G:=Group("C7xD4:5D4"); // GroupNames label
G:=SmallGroup(448,1329);
// by ID
G=gap.SmallGroup(448,1329);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,4790,1690]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations