metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊23D14, C14.1372+ 1+4, C4⋊C4⋊33D14, (C4×D28)⋊13C2, (C4×C28)⋊7C22, C4⋊D28⋊36C2, D14⋊C4⋊7C22, C4.D28⋊8C2, C42⋊2C2⋊2D7, C22⋊D28⋊27C2, D14⋊D4⋊44C2, D28⋊C4⋊39C2, D14⋊Q8⋊40C2, (C2×D28)⋊29C22, C4⋊Dic7⋊61C22, C22⋊C4.40D14, D14.12(C4○D4), D14.D4⋊48C2, D14.5D4⋊38C2, (C2×C28).193C23, (C2×C14).248C24, Dic7⋊C4⋊27C22, C7⋊9(C22.32C24), (C4×Dic7)⋊38C22, C2.62(D4⋊8D14), C23.54(C22×D7), Dic7.D4⋊44C2, (C2×Dic14)⋊11C22, (C22×C14).62C23, (C23×D7).68C22, C22.269(C23×D7), C23.D7.64C22, (C2×Dic7).264C23, (C22×D7).111C23, C2.95(D7×C4○D4), (C2×C4×D7)⋊27C22, C4⋊C4⋊D7⋊41C2, (C7×C4⋊C4)⋊32C22, (D7×C22⋊C4)⋊20C2, (C7×C42⋊2C2)⋊3C2, C14.206(C2×C4○D4), (C2×C4).85(C22×D7), (C2×C7⋊D4).68C22, (C7×C22⋊C4).73C22, SmallGroup(448,1157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊23D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >
Subgroups: 1484 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C42⋊2C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×C7⋊D4, C23×D7, C4×D28, C4.D28, D7×C22⋊C4, C22⋊D28, D14.D4, D14⋊D4, Dic7.D4, D28⋊C4, D14.5D4, C4⋊D28, D14⋊Q8, C4⋊C4⋊D7, C7×C42⋊2C2, C42⋊23D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C23×D7, D7×C4○D4, D4⋊8D14, C42⋊23D14
►On 112 points
Generators in S112
(1 74 13 94)(2 82 14 88)(3 76 8 96)(4 84 9 90)(5 78 10 98)(6 72 11 92)(7 80 12 86)(15 81 22 87)(16 75 23 95)(17 83 24 89)(18 77 25 97)(19 71 26 91)(20 79 27 85)(21 73 28 93)(29 109 36 47)(30 55 37 103)(31 111 38 49)(32 43 39 105)(33 99 40 51)(34 45 41 107)(35 101 42 53)(44 59 106 66)(46 61 108 68)(48 63 110 70)(50 65 112 58)(52 67 100 60)(54 69 102 62)(56 57 104 64)
(1 64 22 38)(2 58 23 32)(3 66 24 40)(4 60 25 34)(5 68 26 42)(6 62 27 36)(7 70 28 30)(8 59 17 33)(9 67 18 41)(10 61 19 35)(11 69 20 29)(12 63 21 37)(13 57 15 31)(14 65 16 39)(43 82 50 95)(44 89 51 76)(45 84 52 97)(46 91 53 78)(47 72 54 85)(48 93 55 80)(49 74 56 87)(71 101 98 108)(73 103 86 110)(75 105 88 112)(77 107 90 100)(79 109 92 102)(81 111 94 104)(83 99 96 106)
(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 10)(11 14)(12 13)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 70)(39 69)(40 68)(41 67)(42 66)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 100)(53 99)(54 112)(55 111)(56 110)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 88)(86 87)(89 98)(90 97)(91 96)(92 95)(93 94)
G:=sub<Sym(112)| (1,74,13,94)(2,82,14,88)(3,76,8,96)(4,84,9,90)(5,78,10,98)(6,72,11,92)(7,80,12,86)(15,81,22,87)(16,75,23,95)(17,83,24,89)(18,77,25,97)(19,71,26,91)(20,79,27,85)(21,73,28,93)(29,109,36,47)(30,55,37,103)(31,111,38,49)(32,43,39,105)(33,99,40,51)(34,45,41,107)(35,101,42,53)(44,59,106,66)(46,61,108,68)(48,63,110,70)(50,65,112,58)(52,67,100,60)(54,69,102,62)(56,57,104,64), (1,64,22,38)(2,58,23,32)(3,66,24,40)(4,60,25,34)(5,68,26,42)(6,62,27,36)(7,70,28,30)(8,59,17,33)(9,67,18,41)(10,61,19,35)(11,69,20,29)(12,63,21,37)(13,57,15,31)(14,65,16,39)(43,82,50,95)(44,89,51,76)(45,84,52,97)(46,91,53,78)(47,72,54,85)(48,93,55,80)(49,74,56,87)(71,101,98,108)(73,103,86,110)(75,105,88,112)(77,107,90,100)(79,109,92,102)(81,111,94,104)(83,99,96,106), (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)(8,10)(11,14)(12,13)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,70)(39,69)(40,68)(41,67)(42,66)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,112)(55,111)(56,110)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,88)(86,87)(89,98)(90,97)(91,96)(92,95)(93,94)>;
G:=Group( (1,74,13,94)(2,82,14,88)(3,76,8,96)(4,84,9,90)(5,78,10,98)(6,72,11,92)(7,80,12,86)(15,81,22,87)(16,75,23,95)(17,83,24,89)(18,77,25,97)(19,71,26,91)(20,79,27,85)(21,73,28,93)(29,109,36,47)(30,55,37,103)(31,111,38,49)(32,43,39,105)(33,99,40,51)(34,45,41,107)(35,101,42,53)(44,59,106,66)(46,61,108,68)(48,63,110,70)(50,65,112,58)(52,67,100,60)(54,69,102,62)(56,57,104,64), (1,64,22,38)(2,58,23,32)(3,66,24,40)(4,60,25,34)(5,68,26,42)(6,62,27,36)(7,70,28,30)(8,59,17,33)(9,67,18,41)(10,61,19,35)(11,69,20,29)(12,63,21,37)(13,57,15,31)(14,65,16,39)(43,82,50,95)(44,89,51,76)(45,84,52,97)(46,91,53,78)(47,72,54,85)(48,93,55,80)(49,74,56,87)(71,101,98,108)(73,103,86,110)(75,105,88,112)(77,107,90,100)(79,109,92,102)(81,111,94,104)(83,99,96,106), (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)(8,10)(11,14)(12,13)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,70)(39,69)(40,68)(41,67)(42,66)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,112)(55,111)(56,110)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,88)(86,87)(89,98)(90,97)(91,96)(92,95)(93,94) );
G=PermutationGroup([[(1,74,13,94),(2,82,14,88),(3,76,8,96),(4,84,9,90),(5,78,10,98),(6,72,11,92),(7,80,12,86),(15,81,22,87),(16,75,23,95),(17,83,24,89),(18,77,25,97),(19,71,26,91),(20,79,27,85),(21,73,28,93),(29,109,36,47),(30,55,37,103),(31,111,38,49),(32,43,39,105),(33,99,40,51),(34,45,41,107),(35,101,42,53),(44,59,106,66),(46,61,108,68),(48,63,110,70),(50,65,112,58),(52,67,100,60),(54,69,102,62),(56,57,104,64)], [(1,64,22,38),(2,58,23,32),(3,66,24,40),(4,60,25,34),(5,68,26,42),(6,62,27,36),(7,70,28,30),(8,59,17,33),(9,67,18,41),(10,61,19,35),(11,69,20,29),(12,63,21,37),(13,57,15,31),(14,65,16,39),(43,82,50,95),(44,89,51,76),(45,84,52,97),(46,91,53,78),(47,72,54,85),(48,93,55,80),(49,74,56,87),(71,101,98,108),(73,103,86,110),(75,105,88,112),(77,107,90,100),(79,109,92,102),(81,111,94,104),(83,99,96,106)], [(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),(8,10),(11,14),(12,13),(15,21),(16,20),(17,19),(22,28),(23,27),(24,26),(29,65),(30,64),(31,63),(32,62),(33,61),(34,60),(35,59),(36,58),(37,57),(38,70),(39,69),(40,68),(41,67),(42,66),(43,109),(44,108),(45,107),(46,106),(47,105),(48,104),(49,103),(50,102),(51,101),(52,100),(53,99),(54,112),(55,111),(56,110),(71,76),(72,75),(73,74),(77,84),(78,83),(79,82),(80,81),(85,88),(86,87),(89,98),(90,97),(91,96),(92,95),(93,94)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 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⋊23D14 | C4×D28 | C4.D28 | D7×C22⋊C4 | C22⋊D28 | D14.D4 | D14⋊D4 | Dic7.D4 | D28⋊C4 | D14.5D4 | C4⋊D28 | D14⋊Q8 | C4⋊C4⋊D7 | 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 | 4 | 3 | 9 | 9 | 2 | 6 | 12 |
Matrix representation of C42⋊23D14 ►in GL6(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 0 | 28 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 8 | 8 | 8 | 8 |
| 0 | 0 | 21 | 3 | 21 | 3 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 8 | 8 | 8 | 8 |
| 0 | 0 | 3 | 21 | 3 | 21 |
G:=sub<GL(6,GF(29))| [17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,27,0,1,0,0,0,0,27,0,1],[28,2,0,0,0,0,28,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,27,18,0,0,0,0,11,2],[1,27,0,0,0,0,0,28,0,0,0,0,0,0,21,8,8,21,0,0,21,26,8,3,0,0,0,0,8,21,0,0,0,0,8,3],[28,2,0,0,0,0,0,1,0,0,0,0,0,0,21,26,8,3,0,0,21,8,8,21,0,0,0,0,8,3,0,0,0,0,8,21] >;
C42⋊23D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{23}D_{14} % in TeX
G:=Group("C4^2:23D14"); // GroupNames label
G:=SmallGroup(448,1157);
// by ID
G=gap.SmallGroup(448,1157);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,192,18822]);
// 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=b^-1,d*c*d=c^-1>;
// generators/relations