metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊24D14, C14.1382+ (1+4), C4⋊C4⋊34D14, (C4×D28)⋊14C2, (C4×C28)⋊8C22, C42⋊D7⋊6C2, C42⋊2C2⋊3D7, D28⋊C4⋊40C2, D14⋊C4⋊63C22, D14⋊2Q8⋊40C2, D14⋊Q8⋊41C2, C22⋊D28.3C2, C4⋊Dic7⋊62C22, C22⋊C4.77D14, D14.26(C4○D4), D14.D4⋊49C2, D14.5D4⋊39C2, (C2×C28).603C23, (C2×C14).249C24, Dic7⋊C4⋊68C22, (C4×Dic7)⋊58C22, C2.63(D4⋊8D14), C23.55(C22×D7), Dic7.D4⋊45C2, C7⋊9(C22.45C24), (C2×Dic14)⋊33C22, (C2×D28).226C22, C23.D14⋊45C2, (C22×C14).63C23, (C23×D7).69C22, C22.270(C23×D7), C23.D7.65C22, (C2×Dic7).129C23, (C22×D7).223C23, C2.96(D7×C4○D4), (C2×C4×D7)⋊53C22, C4⋊C4⋊7D7⋊39C2, (C7×C4⋊C4)⋊33C22, (D7×C22⋊C4)⋊21C2, (C7×C42⋊2C2)⋊4C2, C14.207(C2×C4○D4), (C2×C4).86(C22×D7), (C2×C7⋊D4).69C22, (C7×C22⋊C4).74C22, SmallGroup(448,1158)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1324 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×18], C7, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], D7 [×5], C14 [×3], C14, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×5], C22×C4 [×5], C2×D4 [×3], C2×Q8, C24, Dic7 [×5], C28 [×6], D14 [×4], D14 [×11], C2×C14, C2×C14 [×3], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2, C42⋊2C2, Dic14, C4×D7 [×7], D28 [×4], C2×Dic7 [×5], C7⋊D4, C2×C28 [×6], C22×D7 [×3], C22×D7 [×5], C22×C14, C22.45C24, C4×Dic7 [×2], Dic7⋊C4 [×3], C4⋊Dic7 [×2], D14⋊C4 [×9], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×3], C7×C4⋊C4 [×3], C2×Dic14, C2×C4×D7 [×5], C2×D28 [×2], C2×C7⋊D4, C23×D7, C42⋊D7, C4×D28, C23.D14, D7×C22⋊C4 [×2], C22⋊D28, D14.D4, Dic7.D4, C4⋊C4⋊7D7, D28⋊C4, D14.5D4 [×2], D14⋊Q8, D14⋊2Q8, C7×C42⋊2C2, C42⋊24D14
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, D7×C4○D4 [×2], D4⋊8D14, C42⋊24D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
►On 112 points
(1 99 10 78)(2 107 11 72)(3 101 12 80)(4 109 13 74)(5 103 14 82)(6 111 8 76)(7 105 9 84)(15 102 22 81)(16 110 23 75)(17 104 24 83)(18 112 25 77)(19 106 26 71)(20 100 27 79)(21 108 28 73)(29 65 89 58)(30 45 90 52)(31 67 91 60)(32 47 92 54)(33 69 93 62)(34 49 94 56)(35 57 95 64)(36 51 96 44)(37 59 97 66)(38 53 98 46)(39 61 85 68)(40 55 86 48)(41 63 87 70)(42 43 88 50)
(1 62 26 48)(2 70 27 56)(3 64 28 50)(4 58 22 44)(5 66 23 52)(6 60 24 46)(7 68 25 54)(8 67 17 53)(9 61 18 47)(10 69 19 55)(11 63 20 49)(12 57 21 43)(13 65 15 51)(14 59 16 45)(29 81 36 109)(30 103 37 75)(31 83 38 111)(32 105 39 77)(33 71 40 99)(34 107 41 79)(35 73 42 101)(72 87 100 94)(74 89 102 96)(76 91 104 98)(78 93 106 86)(80 95 108 88)(82 97 110 90)(84 85 112 92)
(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 11)(9 10)(12 14)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 70)(54 69)(55 68)(56 67)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 93)(86 92)(87 91)(88 90)(94 98)(95 97)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)
G:=sub<Sym(112)| (1,99,10,78)(2,107,11,72)(3,101,12,80)(4,109,13,74)(5,103,14,82)(6,111,8,76)(7,105,9,84)(15,102,22,81)(16,110,23,75)(17,104,24,83)(18,112,25,77)(19,106,26,71)(20,100,27,79)(21,108,28,73)(29,65,89,58)(30,45,90,52)(31,67,91,60)(32,47,92,54)(33,69,93,62)(34,49,94,56)(35,57,95,64)(36,51,96,44)(37,59,97,66)(38,53,98,46)(39,61,85,68)(40,55,86,48)(41,63,87,70)(42,43,88,50), (1,62,26,48)(2,70,27,56)(3,64,28,50)(4,58,22,44)(5,66,23,52)(6,60,24,46)(7,68,25,54)(8,67,17,53)(9,61,18,47)(10,69,19,55)(11,63,20,49)(12,57,21,43)(13,65,15,51)(14,59,16,45)(29,81,36,109)(30,103,37,75)(31,83,38,111)(32,105,39,77)(33,71,40,99)(34,107,41,79)(35,73,42,101)(72,87,100,94)(74,89,102,96)(76,91,104,98)(78,93,106,86)(80,95,108,88)(82,97,110,90)(84,85,112,92), (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)(8,11)(9,10)(12,14)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,70)(54,69)(55,68)(56,67)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)>;
G:=Group( (1,99,10,78)(2,107,11,72)(3,101,12,80)(4,109,13,74)(5,103,14,82)(6,111,8,76)(7,105,9,84)(15,102,22,81)(16,110,23,75)(17,104,24,83)(18,112,25,77)(19,106,26,71)(20,100,27,79)(21,108,28,73)(29,65,89,58)(30,45,90,52)(31,67,91,60)(32,47,92,54)(33,69,93,62)(34,49,94,56)(35,57,95,64)(36,51,96,44)(37,59,97,66)(38,53,98,46)(39,61,85,68)(40,55,86,48)(41,63,87,70)(42,43,88,50), (1,62,26,48)(2,70,27,56)(3,64,28,50)(4,58,22,44)(5,66,23,52)(6,60,24,46)(7,68,25,54)(8,67,17,53)(9,61,18,47)(10,69,19,55)(11,63,20,49)(12,57,21,43)(13,65,15,51)(14,59,16,45)(29,81,36,109)(30,103,37,75)(31,83,38,111)(32,105,39,77)(33,71,40,99)(34,107,41,79)(35,73,42,101)(72,87,100,94)(74,89,102,96)(76,91,104,98)(78,93,106,86)(80,95,108,88)(82,97,110,90)(84,85,112,92), (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)(8,11)(9,10)(12,14)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,70)(54,69)(55,68)(56,67)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106) );
G=PermutationGroup([(1,99,10,78),(2,107,11,72),(3,101,12,80),(4,109,13,74),(5,103,14,82),(6,111,8,76),(7,105,9,84),(15,102,22,81),(16,110,23,75),(17,104,24,83),(18,112,25,77),(19,106,26,71),(20,100,27,79),(21,108,28,73),(29,65,89,58),(30,45,90,52),(31,67,91,60),(32,47,92,54),(33,69,93,62),(34,49,94,56),(35,57,95,64),(36,51,96,44),(37,59,97,66),(38,53,98,46),(39,61,85,68),(40,55,86,48),(41,63,87,70),(42,43,88,50)], [(1,62,26,48),(2,70,27,56),(3,64,28,50),(4,58,22,44),(5,66,23,52),(6,60,24,46),(7,68,25,54),(8,67,17,53),(9,61,18,47),(10,69,19,55),(11,63,20,49),(12,57,21,43),(13,65,15,51),(14,59,16,45),(29,81,36,109),(30,103,37,75),(31,83,38,111),(32,105,39,77),(33,71,40,99),(34,107,41,79),(35,73,42,101),(72,87,100,94),(74,89,102,96),(76,91,104,98),(78,93,106,86),(80,95,108,88),(82,97,110,90),(84,85,112,92)], [(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),(8,11),(9,10),(12,14),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,70),(54,69),(55,68),(56,67),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(85,93),(86,92),(87,91),(88,90),(94,98),(95,97),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 25 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 18 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,4,25,0,0,0,0,4,18,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,4,18,0,0,0,0,4,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
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 | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| 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 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2+ (1+4) | D7×C4○D4 | D4⋊8D14 |
| kernel | C42⋊24D14 | C42⋊D7 | C4×D28 | C23.D14 | D7×C22⋊C4 | C22⋊D28 | D14.D4 | Dic7.D4 | C4⋊C4⋊7D7 | D28⋊C4 | D14.5D4 | D14⋊Q8 | D14⋊2Q8 | C7×C42⋊2C2 | C42⋊2C2 | D14 | C42 | C22⋊C4 | C4⋊C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 8 | 3 | 9 | 9 | 1 | 12 | 6 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{24}D_{14} % in TeX
G:=Group("C4^2:24D14"); // GroupNames label
G:=SmallGroup(448,1158);
// by ID
G=gap.SmallGroup(448,1158);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,570,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations