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×C28).713C23, (C2×C14).366C24, (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), C22⋊C4⋊16(C2×C14), (C2×C22⋊C4)⋊14C14, (C14×C22⋊C4)⋊34C2, (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
Subgroups: 570 in 334 conjugacy classes, 166 normal (62 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], C7, C2×C4 [×5], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C23 [×10], C14 [×3], C14 [×9], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24 [×2], C28 [×2], C28 [×8], C2×C14, C2×C14 [×6], C2×C14 [×23], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C2×C28 [×5], C2×C28 [×4], C2×C28 [×10], C7×D4 [×4], C7×D4 [×14], C7×Q8 [×2], C22×C14 [×2], C22×C14 [×4], C22×C14 [×10], D4⋊5D4, C4×C28, C7×C22⋊C4 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×2], C7×C4⋊C4 [×2], C22×C28 [×2], C22×C28 [×4], D4×C14 [×3], D4×C14 [×6], D4×C14 [×4], Q8×C14, C7×C4○D4 [×4], C23×C14 [×2], C14×C22⋊C4 [×2], D4×C28 [×2], C7×C22≀C2 [×2], C7×C4⋊D4, C7×C4⋊D4 [×2], C7×C22⋊Q8, C7×C22.D4 [×2], C7×C4.4D4, D4×C2×C14, C14×C4○D4, C7×D4⋊5D4
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C14 [×35], C22×D4, C2×C4○D4, 2+ (1+4), C7×D4 [×4], C22×C14 [×15], D4⋊5D4, D4×C14 [×6], C7×C4○D4 [×2], C23×C14, D4×C2×C14, C14×C4○D4, C7×2+ (1+4), C7×D4⋊5D4
Generators and relations
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 >
►On 112 points
(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 63 39 75)(2 57 40 76)(3 58 41 77)(4 59 42 71)(5 60 36 72)(6 61 37 73)(7 62 38 74)(8 93 23 84)(9 94 24 78)(10 95 25 79)(11 96 26 80)(12 97 27 81)(13 98 28 82)(14 92 22 83)(15 87 109 99)(16 88 110 100)(17 89 111 101)(18 90 112 102)(19 91 106 103)(20 85 107 104)(21 86 108 105)(29 67 48 51)(30 68 49 52)(31 69 43 53)(32 70 44 54)(33 64 45 55)(34 65 46 56)(35 66 47 50)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 43)(50 63)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 101)(93 102)(94 103)(95 104)(96 105)(97 99)(98 100)
(1 94 35 91)(2 95 29 85)(3 96 30 86)(4 97 31 87)(5 98 32 88)(6 92 33 89)(7 93 34 90)(8 56 18 74)(9 50 19 75)(10 51 20 76)(11 52 21 77)(12 53 15 71)(13 54 16 72)(14 55 17 73)(22 64 111 61)(23 65 112 62)(24 66 106 63)(25 67 107 57)(26 68 108 58)(27 69 109 59)(28 70 110 60)(36 82 44 100)(37 83 45 101)(38 84 46 102)(39 78 47 103)(40 79 48 104)(41 80 49 105)(42 81 43 99)
(1 91)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 56)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 95)(30 96)(31 97)(32 98)(33 92)(34 93)(35 94)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 99)(43 81)(44 82)(45 83)(46 84)(47 78)(48 79)(49 80)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 106)
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), (1,63,39,75)(2,57,40,76)(3,58,41,77)(4,59,42,71)(5,60,36,72)(6,61,37,73)(7,62,38,74)(8,93,23,84)(9,94,24,78)(10,95,25,79)(11,96,26,80)(12,97,27,81)(13,98,28,82)(14,92,22,83)(15,87,109,99)(16,88,110,100)(17,89,111,101)(18,90,112,102)(19,91,106,103)(20,85,107,104)(21,86,108,105)(29,67,48,51)(30,68,49,52)(31,69,43,53)(32,70,44,54)(33,64,45,55)(34,65,46,56)(35,66,47,50), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,63)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,56,18,74)(9,50,19,75)(10,51,20,76)(11,52,21,77)(12,53,15,71)(13,54,16,72)(14,55,17,73)(22,64,111,61)(23,65,112,62)(24,66,106,63)(25,67,107,57)(26,68,108,58)(27,69,109,59)(28,70,110,60)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99), (1,91)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,56)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,95)(30,96)(31,97)(32,98)(33,92)(34,93)(35,94)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,99)(43,81)(44,82)(45,83)(46,84)(47,78)(48,79)(49,80)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,106)>;
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), (1,63,39,75)(2,57,40,76)(3,58,41,77)(4,59,42,71)(5,60,36,72)(6,61,37,73)(7,62,38,74)(8,93,23,84)(9,94,24,78)(10,95,25,79)(11,96,26,80)(12,97,27,81)(13,98,28,82)(14,92,22,83)(15,87,109,99)(16,88,110,100)(17,89,111,101)(18,90,112,102)(19,91,106,103)(20,85,107,104)(21,86,108,105)(29,67,48,51)(30,68,49,52)(31,69,43,53)(32,70,44,54)(33,64,45,55)(34,65,46,56)(35,66,47,50), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,63)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,56,18,74)(9,50,19,75)(10,51,20,76)(11,52,21,77)(12,53,15,71)(13,54,16,72)(14,55,17,73)(22,64,111,61)(23,65,112,62)(24,66,106,63)(25,67,107,57)(26,68,108,58)(27,69,109,59)(28,70,110,60)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99), (1,91)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,56)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,95)(30,96)(31,97)(32,98)(33,92)(34,93)(35,94)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,99)(43,81)(44,82)(45,83)(46,84)(47,78)(48,79)(49,80)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,106) );
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)], [(1,63,39,75),(2,57,40,76),(3,58,41,77),(4,59,42,71),(5,60,36,72),(6,61,37,73),(7,62,38,74),(8,93,23,84),(9,94,24,78),(10,95,25,79),(11,96,26,80),(12,97,27,81),(13,98,28,82),(14,92,22,83),(15,87,109,99),(16,88,110,100),(17,89,111,101),(18,90,112,102),(19,91,106,103),(20,85,107,104),(21,86,108,105),(29,67,48,51),(30,68,49,52),(31,69,43,53),(32,70,44,54),(33,64,45,55),(34,65,46,56),(35,66,47,50)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(36,44),(37,45),(38,46),(39,47),(40,48),(41,49),(42,43),(50,63),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62),(64,73),(65,74),(66,75),(67,76),(68,77),(69,71),(70,72),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,101),(93,102),(94,103),(95,104),(96,105),(97,99),(98,100)], [(1,94,35,91),(2,95,29,85),(3,96,30,86),(4,97,31,87),(5,98,32,88),(6,92,33,89),(7,93,34,90),(8,56,18,74),(9,50,19,75),(10,51,20,76),(11,52,21,77),(12,53,15,71),(13,54,16,72),(14,55,17,73),(22,64,111,61),(23,65,112,62),(24,66,106,63),(25,67,107,57),(26,68,108,58),(27,69,109,59),(28,70,110,60),(36,82,44,100),(37,83,45,101),(38,84,46,102),(39,78,47,103),(40,79,48,104),(41,80,49,105),(42,81,43,99)], [(1,91),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,56),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,95),(30,96),(31,97),(32,98),(33,92),(34,93),(35,94),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,99),(43,81),(44,82),(45,83),(46,84),(47,78),(48,79),(49,80),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,106)])
Matrix representation ►G ⊆ 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] >;
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 |
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