direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C8○D4, D4.C28, Q8.C28, M4(2)⋊5C14, C28.54C23, C56.30C22, C56○(C7×D4), C56○(C7×Q8), (C2×C8)⋊7C14, (C2×C56)⋊15C2, C4.5(C2×C28), C8.7(C2×C14), (C7×D4).2C4, (C7×Q8).2C4, C28.32(C2×C4), C4○D4.3C14, C56○(C7×M4(2)), C2.7(C22×C28), C22.1(C2×C28), (C7×M4(2))⋊11C2, C14.35(C22×C4), C4.12(C22×C14), (C2×C28).128C22, C56○(C7×C4○D4), (C2×C14).8(C2×C4), (C7×C4○D4).6C2, (C2×C4).24(C2×C14), SmallGroup(224,166)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8○D4
G = < a,b,c,d | a7=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C7, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C14, C14 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C28, C28 [×3], C2×C14 [×3], C8○D4, C56, C56 [×3], C2×C28 [×3], C7×D4 [×3], C7×Q8, C2×C56 [×3], C7×M4(2) [×3], C7×C4○D4, C7×C8○D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C7, C2×C4 [×6], C23, C14 [×7], C22×C4, C28 [×4], C2×C14 [×7], C8○D4, C2×C28 [×6], C22×C14, C22×C28, C7×C8○D4
►On 112 points
Generators in S112
(1 31 47 79 95 23 39)(2 32 48 80 96 24 40)(3 25 41 73 89 17 33)(4 26 42 74 90 18 34)(5 27 43 75 91 19 35)(6 28 44 76 92 20 36)(7 29 45 77 93 21 37)(8 30 46 78 94 22 38)(9 106 66 50 82 98 62)(10 107 67 51 83 99 63)(11 108 68 52 84 100 64)(12 109 69 53 85 101 57)(13 110 70 54 86 102 58)(14 111 71 55 87 103 59)(15 112 72 56 88 104 60)(16 105 65 49 81 97 61)
(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 99 5 103)(2 100 6 104)(3 101 7 97)(4 102 8 98)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 41)(17 53 21 49)(18 54 22 50)(19 55 23 51)(20 56 24 52)(25 57 29 61)(26 58 30 62)(27 59 31 63)(28 60 32 64)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)(65 89 69 93)(66 90 70 94)(67 91 71 95)(68 92 72 96)(73 109 77 105)(74 110 78 106)(75 111 79 107)(76 112 80 108)
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)
G:=sub<Sym(112)| (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,106,66,50,82,98,62)(10,107,67,51,83,99,63)(11,108,68,52,84,100,64)(12,109,69,53,85,101,57)(13,110,70,54,86,102,58)(14,111,71,55,87,103,59)(15,112,72,56,88,104,60)(16,105,65,49,81,97,61), (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), (1,99,5,103)(2,100,6,104)(3,101,7,97)(4,102,8,98)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(17,53,21,49)(18,54,22,50)(19,55,23,51)(20,56,24,52)(25,57,29,61)(26,58,30,62)(27,59,31,63)(28,60,32,64)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)(65,89,69,93)(66,90,70,94)(67,91,71,95)(68,92,72,96)(73,109,77,105)(74,110,78,106)(75,111,79,107)(76,112,80,108), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)>;
G:=Group( (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,106,66,50,82,98,62)(10,107,67,51,83,99,63)(11,108,68,52,84,100,64)(12,109,69,53,85,101,57)(13,110,70,54,86,102,58)(14,111,71,55,87,103,59)(15,112,72,56,88,104,60)(16,105,65,49,81,97,61), (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), (1,99,5,103)(2,100,6,104)(3,101,7,97)(4,102,8,98)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(17,53,21,49)(18,54,22,50)(19,55,23,51)(20,56,24,52)(25,57,29,61)(26,58,30,62)(27,59,31,63)(28,60,32,64)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)(65,89,69,93)(66,90,70,94)(67,91,71,95)(68,92,72,96)(73,109,77,105)(74,110,78,106)(75,111,79,107)(76,112,80,108), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112) );
G=PermutationGroup([(1,31,47,79,95,23,39),(2,32,48,80,96,24,40),(3,25,41,73,89,17,33),(4,26,42,74,90,18,34),(5,27,43,75,91,19,35),(6,28,44,76,92,20,36),(7,29,45,77,93,21,37),(8,30,46,78,94,22,38),(9,106,66,50,82,98,62),(10,107,67,51,83,99,63),(11,108,68,52,84,100,64),(12,109,69,53,85,101,57),(13,110,70,54,86,102,58),(14,111,71,55,87,103,59),(15,112,72,56,88,104,60),(16,105,65,49,81,97,61)], [(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)], [(1,99,5,103),(2,100,6,104),(3,101,7,97),(4,102,8,98),(9,42,13,46),(10,43,14,47),(11,44,15,48),(12,45,16,41),(17,53,21,49),(18,54,22,50),(19,55,23,51),(20,56,24,52),(25,57,29,61),(26,58,30,62),(27,59,31,63),(28,60,32,64),(33,85,37,81),(34,86,38,82),(35,87,39,83),(36,88,40,84),(65,89,69,93),(66,90,70,94),(67,91,71,95),(68,92,72,96),(73,109,77,105),(74,110,78,106),(75,111,79,107),(76,112,80,108)], [(1,103),(2,104),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,61),(26,62),(27,63),(28,64),(29,57),(30,58),(31,59),(32,60),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112)])
C7×C8○D4 is a maximal subgroup of
C56.92D4 C56.70C23 D4.3D28 D4.4D28 D4.5D28 C56.93D4 C56.50D4 C56.49C23 D4.11D28 D4.12D28 D4.13D28
C7×C8○D4 is a maximal quotient of
D4×C56 Q8×C56
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 14A | ··· | 14F | 14G | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28AD | 56A | ··· | 56X | 56Y | ··· | 56BH |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | C8○D4 | C7×C8○D4 |
| kernel | C7×C8○D4 | C2×C56 | C7×M4(2) | C7×C4○D4 | C7×D4 | C7×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 6 | 18 | 18 | 6 | 36 | 12 | 4 | 24 |
Matrix representation of C7×C8○D4 ►in GL3(𝔽113) generated by
| 106 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 44 | 0 |
| 0 | 0 | 44 |
| 1 | 0 | 0 |
| 0 | 101 | 111 |
| 0 | 16 | 12 |
| 112 | 0 | 0 |
| 0 | 101 | 111 |
| 0 | 15 | 12 |
G:=sub<GL(3,GF(113))| [106,0,0,0,1,0,0,0,1],[1,0,0,0,44,0,0,0,44],[1,0,0,0,101,16,0,111,12],[112,0,0,0,101,15,0,111,12] >;
C7×C8○D4 in GAP, Magma, Sage, TeX
C_7\times C_8\circ D_4
% in TeX
G:=Group("C7xC8oD4"); // GroupNames label
G:=SmallGroup(224,166);
// by ID
G=gap.SmallGroup(224,166);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,1052,88]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations