direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×Q8○M4(2), C56.81C23, C28.94C24, C8○D4⋊8C14, C4○D4.2C28, D4.9(C2×C28), (C2×Q8).8C28, (C2×C56)⋊39C22, (C2×D4).10C28, (D4×C14).22C4, (Q8×C14).18C4, Q8.10(C2×C28), C23.13(C2×C28), C4.23(C22×C28), C2.12(C23×C28), C8.14(C22×C14), C14.64(C23×C4), C4.18(C23×C14), (C2×M4(2))⋊16C14, (C14×M4(2))⋊34C2, M4(2)⋊12(C2×C14), (C2×C28).970C23, C28.168(C22×C4), C22.5(C22×C28), (C7×M4(2))⋊41C22, (C22×C28).464C22, (C2×C8)⋊9(C2×C14), (C7×C8○D4)⋊17C2, (C7×C4○D4).6C4, (C2×C4).32(C2×C28), (C7×D4).31(C2×C4), (C7×Q8).34(C2×C4), (C2×C28).204(C2×C4), C4○D4.15(C2×C14), (C14×C4○D4).25C2, (C2×C4○D4).11C14, (C22×C14).25(C2×C4), (C22×C4).75(C2×C14), (C2×C14).36(C22×C4), (C7×C4○D4).60C22, (C2×C4).140(C22×C14), SmallGroup(448,1351)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 290 in 258 conjugacy classes, 238 normal (18 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C7, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C14, C14 [×7], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C28 [×2], C28 [×6], C2×C14, C2×C14 [×6], C2×C14 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C56 [×8], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×C14 [×3], Q8○M4(2), C2×C56 [×12], C7×M4(2) [×16], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C14×M4(2) [×6], C7×C8○D4 [×8], C14×C4○D4, C7×Q8○M4(2)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C7, C2×C4 [×28], C23 [×15], C14 [×15], C22×C4 [×14], C24, C28 [×8], C2×C14 [×35], C23×C4, C2×C28 [×28], C22×C14 [×15], Q8○M4(2), C22×C28 [×14], C23×C14, C23×C28, C7×Q8○M4(2)
Generators and relations
G = < a,b,c,d,e | a7=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►On 112 points
Generators in S112
(1 26 42 75 91 19 34)(2 27 43 76 92 20 35)(3 28 44 77 93 21 36)(4 29 45 78 94 22 37)(5 30 46 79 95 23 38)(6 31 47 80 96 24 39)(7 32 48 73 89 17 40)(8 25 41 74 90 18 33)(9 49 82 98 61 68 106)(10 50 83 99 62 69 107)(11 51 84 100 63 70 108)(12 52 85 101 64 71 109)(13 53 86 102 57 72 110)(14 54 87 103 58 65 111)(15 55 88 104 59 66 112)(16 56 81 97 60 67 105)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 75 77 79)(74 76 78 80)(81 87 85 83)(82 88 86 84)(89 91 93 95)(90 92 94 96)(97 103 101 99)(98 104 102 100)(105 111 109 107)(106 112 110 108)
(1 99 5 103)(2 100 6 104)(3 101 7 97)(4 102 8 98)(9 94 13 90)(10 95 14 91)(11 96 15 92)(12 89 16 93)(17 56 21 52)(18 49 22 53)(19 50 23 54)(20 51 24 55)(25 61 29 57)(26 62 30 58)(27 63 31 59)(28 64 32 60)(33 82 37 86)(34 83 38 87)(35 84 39 88)(36 85 40 81)(41 68 45 72)(42 69 46 65)(43 70 47 66)(44 71 48 67)(73 105 77 109)(74 106 78 110)(75 107 79 111)(76 108 80 112)
(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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(106 110)(108 112)
G:=sub<Sym(112)| (1,26,42,75,91,19,34)(2,27,43,76,92,20,35)(3,28,44,77,93,21,36)(4,29,45,78,94,22,37)(5,30,46,79,95,23,38)(6,31,47,80,96,24,39)(7,32,48,73,89,17,40)(8,25,41,74,90,18,33)(9,49,82,98,61,68,106)(10,50,83,99,62,69,107)(11,51,84,100,63,70,108)(12,52,85,101,64,71,109)(13,53,86,102,57,72,110)(14,54,87,103,58,65,111)(15,55,88,104,59,66,112)(16,56,81,97,60,67,105), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,103,101,99)(98,104,102,100)(105,111,109,107)(106,112,110,108), (1,99,5,103)(2,100,6,104)(3,101,7,97)(4,102,8,98)(9,94,13,90)(10,95,14,91)(11,96,15,92)(12,89,16,93)(17,56,21,52)(18,49,22,53)(19,50,23,54)(20,51,24,55)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60)(33,82,37,86)(34,83,38,87)(35,84,39,88)(36,85,40,81)(41,68,45,72)(42,69,46,65)(43,70,47,66)(44,71,48,67)(73,105,77,109)(74,106,78,110)(75,107,79,111)(76,108,80,112), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(98,102)(100,104)(106,110)(108,112)>;
G:=Group( (1,26,42,75,91,19,34)(2,27,43,76,92,20,35)(3,28,44,77,93,21,36)(4,29,45,78,94,22,37)(5,30,46,79,95,23,38)(6,31,47,80,96,24,39)(7,32,48,73,89,17,40)(8,25,41,74,90,18,33)(9,49,82,98,61,68,106)(10,50,83,99,62,69,107)(11,51,84,100,63,70,108)(12,52,85,101,64,71,109)(13,53,86,102,57,72,110)(14,54,87,103,58,65,111)(15,55,88,104,59,66,112)(16,56,81,97,60,67,105), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,103,101,99)(98,104,102,100)(105,111,109,107)(106,112,110,108), (1,99,5,103)(2,100,6,104)(3,101,7,97)(4,102,8,98)(9,94,13,90)(10,95,14,91)(11,96,15,92)(12,89,16,93)(17,56,21,52)(18,49,22,53)(19,50,23,54)(20,51,24,55)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60)(33,82,37,86)(34,83,38,87)(35,84,39,88)(36,85,40,81)(41,68,45,72)(42,69,46,65)(43,70,47,66)(44,71,48,67)(73,105,77,109)(74,106,78,110)(75,107,79,111)(76,108,80,112), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(98,102)(100,104)(106,110)(108,112) );
G=PermutationGroup([(1,26,42,75,91,19,34),(2,27,43,76,92,20,35),(3,28,44,77,93,21,36),(4,29,45,78,94,22,37),(5,30,46,79,95,23,38),(6,31,47,80,96,24,39),(7,32,48,73,89,17,40),(8,25,41,74,90,18,33),(9,49,82,98,61,68,106),(10,50,83,99,62,69,107),(11,51,84,100,63,70,108),(12,52,85,101,64,71,109),(13,53,86,102,57,72,110),(14,54,87,103,58,65,111),(15,55,88,104,59,66,112),(16,56,81,97,60,67,105)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,75,77,79),(74,76,78,80),(81,87,85,83),(82,88,86,84),(89,91,93,95),(90,92,94,96),(97,103,101,99),(98,104,102,100),(105,111,109,107),(106,112,110,108)], [(1,99,5,103),(2,100,6,104),(3,101,7,97),(4,102,8,98),(9,94,13,90),(10,95,14,91),(11,96,15,92),(12,89,16,93),(17,56,21,52),(18,49,22,53),(19,50,23,54),(20,51,24,55),(25,61,29,57),(26,62,30,58),(27,63,31,59),(28,64,32,60),(33,82,37,86),(34,83,38,87),(35,84,39,88),(36,85,40,81),(41,68,45,72),(42,69,46,65),(43,70,47,66),(44,71,48,67),(73,105,77,109),(74,106,78,110),(75,107,79,111),(76,108,80,112)], [(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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(66,70),(68,72),(74,78),(76,80),(82,86),(84,88),(90,94),(92,96),(98,102),(100,104),(106,110),(108,112)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 15 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 98 | 15 | 0 |
| 95 | 0 | 0 | 98 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 78 | 47 | 0 | 112 |
| 66 | 78 | 1 | 0 |
| 94 | 1 | 111 | 0 |
| 69 | 0 | 0 | 2 |
| 38 | 47 | 19 | 1 |
| 34 | 86 | 69 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 94 | 1 | 112 | 0 |
| 44 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[15,0,0,95,0,98,98,0,0,0,15,0,0,0,0,98],[0,112,78,66,1,0,47,78,0,0,0,1,0,0,112,0],[94,69,38,34,1,0,47,86,111,0,19,69,0,2,1,0],[1,0,94,44,0,1,1,0,0,0,112,0,0,0,0,112] >;
238 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 4A | 4B | 4C | ··· | 4I | 7A | ··· | 7F | 8A | ··· | 8P | 14A | ··· | 14F | 14G | ··· | 14AV | 28A | ··· | 28L | 28M | ··· | 28BB | 56A | ··· | 56CR |
| order | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
238 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | C28 | Q8○M4(2) | C7×Q8○M4(2) |
| kernel | C7×Q8○M4(2) | C14×M4(2) | C7×C8○D4 | C14×C4○D4 | D4×C14 | Q8×C14 | C7×C4○D4 | Q8○M4(2) | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C7 | C1 |
| # reps | 1 | 6 | 8 | 1 | 6 | 2 | 8 | 6 | 36 | 48 | 6 | 36 | 12 | 48 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_7\times Q_8\circ M_{4(2)} % in TeX
G:=Group("C7xQ8oM4(2)"); // GroupNames label
G:=SmallGroup(448,1351);
// by ID
G=gap.SmallGroup(448,1351);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,2403,6499,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀