metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.3Dic7, C4⋊Q8.3D7, (C4×C28).3C4, (C2×C28).6D4, (Q8×C14).3C4, (C2×Q8).4D14, (C2×Q8).3Dic7, C7⋊2(C42.3C4), (Q8×C14).4C22, C14.26(C23⋊C4), C28.10D4.2C2, C2.11(C23⋊Dic7), C22.17(C23.D7), (C7×C4⋊Q8).3C2, (C2×C28).11(C2×C4), (C2×C4).8(C7⋊D4), (C2×C4).4(C2×Dic7), (C2×C14).106(C22⋊C4), SmallGroup(448,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.3Dic7
G = < a,b,c,d | a4=b4=1, c14=b2, d2=b2c7, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c13 >
Subgroups: 204 in 60 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, M4(2), C2×Q8, C28, C2×C14, C4.10D4, C4⋊Q8, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×Q8, C42.3C4, C4.Dic7, C4×C28, C7×C4⋊C4, Q8×C14, C28.10D4, C7×C4⋊Q8, C42.3Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C42.3C4, C23.D7, C23⋊Dic7, C42.3Dic7
►On 112 points
Generators in S112
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 102 71 88)(58 89 72 103)(59 104 73 90)(60 91 74 105)(61 106 75 92)(62 93 76 107)(63 108 77 94)(64 95 78 109)(65 110 79 96)(66 97 80 111)(67 112 81 98)(68 99 82 85)(69 86 83 100)(70 101 84 87)
(1 43 15 29)(2 30 16 44)(3 45 17 31)(4 32 18 46)(5 47 19 33)(6 34 20 48)(7 49 21 35)(8 36 22 50)(9 51 23 37)(10 38 24 52)(11 53 25 39)(12 40 26 54)(13 55 27 41)(14 42 28 56)(57 88 71 102)(58 103 72 89)(59 90 73 104)(60 105 74 91)(61 92 75 106)(62 107 76 93)(63 94 77 108)(64 109 78 95)(65 96 79 110)(66 111 80 97)(67 98 81 112)(68 85 82 99)(69 100 83 86)(70 87 84 101)
(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 95 22 88 15 109 8 102)(2 108 23 101 16 94 9 87)(3 93 24 86 17 107 10 100)(4 106 25 99 18 92 11 85)(5 91 26 112 19 105 12 98)(6 104 27 97 20 90 13 111)(7 89 28 110 21 103 14 96)(29 78 50 71 43 64 36 57)(30 63 51 84 44 77 37 70)(31 76 52 69 45 62 38 83)(32 61 53 82 46 75 39 68)(33 74 54 67 47 60 40 81)(34 59 55 80 48 73 41 66)(35 72 56 65 49 58 42 79)
G:=sub<Sym(112)| (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,102,71,88)(58,89,72,103)(59,104,73,90)(60,91,74,105)(61,106,75,92)(62,93,76,107)(63,108,77,94)(64,95,78,109)(65,110,79,96)(66,97,80,111)(67,112,81,98)(68,99,82,85)(69,86,83,100)(70,101,84,87), (1,43,15,29)(2,30,16,44)(3,45,17,31)(4,32,18,46)(5,47,19,33)(6,34,20,48)(7,49,21,35)(8,36,22,50)(9,51,23,37)(10,38,24,52)(11,53,25,39)(12,40,26,54)(13,55,27,41)(14,42,28,56)(57,88,71,102)(58,103,72,89)(59,90,73,104)(60,105,74,91)(61,92,75,106)(62,107,76,93)(63,94,77,108)(64,109,78,95)(65,96,79,110)(66,111,80,97)(67,98,81,112)(68,85,82,99)(69,100,83,86)(70,87,84,101), (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), (1,95,22,88,15,109,8,102)(2,108,23,101,16,94,9,87)(3,93,24,86,17,107,10,100)(4,106,25,99,18,92,11,85)(5,91,26,112,19,105,12,98)(6,104,27,97,20,90,13,111)(7,89,28,110,21,103,14,96)(29,78,50,71,43,64,36,57)(30,63,51,84,44,77,37,70)(31,76,52,69,45,62,38,83)(32,61,53,82,46,75,39,68)(33,74,54,67,47,60,40,81)(34,59,55,80,48,73,41,66)(35,72,56,65,49,58,42,79)>;
G:=Group( (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,102,71,88)(58,89,72,103)(59,104,73,90)(60,91,74,105)(61,106,75,92)(62,93,76,107)(63,108,77,94)(64,95,78,109)(65,110,79,96)(66,97,80,111)(67,112,81,98)(68,99,82,85)(69,86,83,100)(70,101,84,87), (1,43,15,29)(2,30,16,44)(3,45,17,31)(4,32,18,46)(5,47,19,33)(6,34,20,48)(7,49,21,35)(8,36,22,50)(9,51,23,37)(10,38,24,52)(11,53,25,39)(12,40,26,54)(13,55,27,41)(14,42,28,56)(57,88,71,102)(58,103,72,89)(59,90,73,104)(60,105,74,91)(61,92,75,106)(62,107,76,93)(63,94,77,108)(64,109,78,95)(65,96,79,110)(66,111,80,97)(67,98,81,112)(68,85,82,99)(69,100,83,86)(70,87,84,101), (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), (1,95,22,88,15,109,8,102)(2,108,23,101,16,94,9,87)(3,93,24,86,17,107,10,100)(4,106,25,99,18,92,11,85)(5,91,26,112,19,105,12,98)(6,104,27,97,20,90,13,111)(7,89,28,110,21,103,14,96)(29,78,50,71,43,64,36,57)(30,63,51,84,44,77,37,70)(31,76,52,69,45,62,38,83)(32,61,53,82,46,75,39,68)(33,74,54,67,47,60,40,81)(34,59,55,80,48,73,41,66)(35,72,56,65,49,58,42,79) );
G=PermutationGroup([[(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,102,71,88),(58,89,72,103),(59,104,73,90),(60,91,74,105),(61,106,75,92),(62,93,76,107),(63,108,77,94),(64,95,78,109),(65,110,79,96),(66,97,80,111),(67,112,81,98),(68,99,82,85),(69,86,83,100),(70,101,84,87)], [(1,43,15,29),(2,30,16,44),(3,45,17,31),(4,32,18,46),(5,47,19,33),(6,34,20,48),(7,49,21,35),(8,36,22,50),(9,51,23,37),(10,38,24,52),(11,53,25,39),(12,40,26,54),(13,55,27,41),(14,42,28,56),(57,88,71,102),(58,103,72,89),(59,90,73,104),(60,105,74,91),(61,92,75,106),(62,107,76,93),(63,94,77,108),(64,109,78,95),(65,96,79,110),(66,111,80,97),(67,98,81,112),(68,85,82,99),(69,100,83,86),(70,87,84,101)], [(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)], [(1,95,22,88,15,109,8,102),(2,108,23,101,16,94,9,87),(3,93,24,86,17,107,10,100),(4,106,25,99,18,92,11,85),(5,91,26,112,19,105,12,98),(6,104,27,97,20,90,13,111),(7,89,28,110,21,103,14,96),(29,78,50,71,43,64,36,57),(30,63,51,84,44,77,37,70),(31,76,52,69,45,62,38,83),(32,61,53,82,46,75,39,68),(33,74,54,67,47,60,40,81),(34,59,55,80,48,73,41,66),(35,72,56,65,49,58,42,79)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 4A | ··· | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 4 | ··· | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | ··· | 4 | 8 | 2 | 2 | 2 | 56 | 56 | 56 | 56 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | - | |||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D7 | Dic7 | Dic7 | D14 | C7⋊D4 | C23⋊C4 | C42.3C4 | C23⋊Dic7 | C42.3Dic7 |
| kernel | C42.3Dic7 | C28.10D4 | C7×C4⋊Q8 | C4×C28 | Q8×C14 | C2×C28 | C4⋊Q8 | C42 | C2×Q8 | C2×Q8 | C2×C4 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of C42.3Dic7 ►in GL4(𝔽113) generated by
| 112 | 0 | 96 | 48 |
| 0 | 112 | 14 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 37 | 112 |
| 112 | 37 | 49 | 85 |
| 3 | 1 | 51 | 62 |
| 0 | 0 | 112 | 110 |
| 0 | 0 | 76 | 1 |
| 51 | 8 | 13 | 70 |
| 109 | 62 | 91 | 29 |
| 0 | 0 | 33 | 69 |
| 0 | 0 | 31 | 80 |
| 97 | 56 | 105 | 91 |
| 60 | 87 | 57 | 74 |
| 111 | 19 | 32 | 94 |
| 94 | 12 | 78 | 10 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,96,14,1,37,48,0,3,112],[112,3,0,0,37,1,0,0,49,51,112,76,85,62,110,1],[51,109,0,0,8,62,0,0,13,91,33,31,70,29,69,80],[97,60,111,94,56,87,19,12,105,57,32,78,91,74,94,10] >;
C42.3Dic7 in GAP, Magma, Sage, TeX
C_4^2._3{\rm Dic}_7 % in TeX
G:=Group("C4^2.3Dic7"); // GroupNames label
G:=SmallGroup(448,105);
// by ID
G=gap.SmallGroup(448,105);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,219,184,1571,570,297,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^13>;
// generators/relations