direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×M4(2).C4, M4(2).1C28, C8.4(C2×C28), C56.45(C2×C4), C4.78(D4×C14), C28.65(C4⋊C4), (C2×C28).44Q8, C8.C4⋊3C14, (C2×C28).523D4, C28.483(C2×D4), C23.5(C7×Q8), C22.2(Q8×C14), (C22×C14).5Q8, C4.29(C22×C28), (C7×M4(2)).3C4, C28.187(C22×C4), (C2×C56).270C22, (C2×C28).903C23, (C2×M4(2)).2C14, (C14×M4(2)).34C2, M4(2).10(C2×C14), (C22×C28).415C22, (C7×M4(2)).44C22, C4.16(C7×C4⋊C4), C2.16(C14×C4⋊C4), C14.72(C2×C4⋊C4), (C2×C4).7(C7×Q8), (C2×C8).17(C2×C14), (C2×C4).26(C2×C28), (C2×C4).126(C7×D4), C22.12(C7×C4⋊C4), (C2×C14).15(C2×Q8), (C7×C8.C4)⋊12C2, (C2×C14).28(C4⋊C4), (C2×C28).199(C2×C4), (C2×C4).78(C22×C14), (C22×C4).34(C2×C14), SmallGroup(448,838)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×M4(2).C4
G = < a,b,c,d | a7=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >
Subgroups: 130 in 102 conjugacy classes, 78 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C8.C4, C2×M4(2), C2×M4(2), C56, C56, C2×C28, C2×C28, C22×C14, M4(2).C4, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C7×C8.C4, C14×M4(2), C14×M4(2), C7×M4(2).C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2×C4⋊C4, C2×C28, C7×D4, C7×Q8, C22×C14, M4(2).C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C14×C4⋊C4, C7×M4(2).C4
►On 112 points
(1 61 18 49 39 41 10)(2 62 19 50 40 42 11)(3 63 20 51 33 43 12)(4 64 21 52 34 44 13)(5 57 22 53 35 45 14)(6 58 23 54 36 46 15)(7 59 24 55 37 47 16)(8 60 17 56 38 48 9)(25 79 108 91 88 71 100)(26 80 109 92 81 72 101)(27 73 110 93 82 65 102)(28 74 111 94 83 66 103)(29 75 112 95 84 67 104)(30 76 105 96 85 68 97)(31 77 106 89 86 69 98)(32 78 107 90 87 70 99)
(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 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)(82 86)(84 88)(89 93)(91 95)(98 102)(100 104)(106 110)(108 112)
(1 97 7 99 5 101 3 103)(2 104 8 98 6 100 4 102)(9 69 15 71 13 65 11 67)(10 68 16 70 14 72 12 66)(17 77 23 79 21 73 19 75)(18 76 24 78 22 80 20 74)(25 64 27 62 29 60 31 58)(26 63 28 61 30 59 32 57)(33 94 39 96 37 90 35 92)(34 93 40 95 38 89 36 91)(41 85 47 87 45 81 43 83)(42 84 48 86 46 88 44 82)(49 105 55 107 53 109 51 111)(50 112 56 106 54 108 52 110)
G:=sub<Sym(112)| (1,61,18,49,39,41,10)(2,62,19,50,40,42,11)(3,63,20,51,33,43,12)(4,64,21,52,34,44,13)(5,57,22,53,35,45,14)(6,58,23,54,36,46,15)(7,59,24,55,37,47,16)(8,60,17,56,38,48,9)(25,79,108,91,88,71,100)(26,80,109,92,81,72,101)(27,73,110,93,82,65,102)(28,74,111,94,83,66,103)(29,75,112,95,84,67,104)(30,76,105,96,85,68,97)(31,77,106,89,86,69,98)(32,78,107,90,87,70,99), (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), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79)(82,86)(84,88)(89,93)(91,95)(98,102)(100,104)(106,110)(108,112), (1,97,7,99,5,101,3,103)(2,104,8,98,6,100,4,102)(9,69,15,71,13,65,11,67)(10,68,16,70,14,72,12,66)(17,77,23,79,21,73,19,75)(18,76,24,78,22,80,20,74)(25,64,27,62,29,60,31,58)(26,63,28,61,30,59,32,57)(33,94,39,96,37,90,35,92)(34,93,40,95,38,89,36,91)(41,85,47,87,45,81,43,83)(42,84,48,86,46,88,44,82)(49,105,55,107,53,109,51,111)(50,112,56,106,54,108,52,110)>;
G:=Group( (1,61,18,49,39,41,10)(2,62,19,50,40,42,11)(3,63,20,51,33,43,12)(4,64,21,52,34,44,13)(5,57,22,53,35,45,14)(6,58,23,54,36,46,15)(7,59,24,55,37,47,16)(8,60,17,56,38,48,9)(25,79,108,91,88,71,100)(26,80,109,92,81,72,101)(27,73,110,93,82,65,102)(28,74,111,94,83,66,103)(29,75,112,95,84,67,104)(30,76,105,96,85,68,97)(31,77,106,89,86,69,98)(32,78,107,90,87,70,99), (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), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79)(82,86)(84,88)(89,93)(91,95)(98,102)(100,104)(106,110)(108,112), (1,97,7,99,5,101,3,103)(2,104,8,98,6,100,4,102)(9,69,15,71,13,65,11,67)(10,68,16,70,14,72,12,66)(17,77,23,79,21,73,19,75)(18,76,24,78,22,80,20,74)(25,64,27,62,29,60,31,58)(26,63,28,61,30,59,32,57)(33,94,39,96,37,90,35,92)(34,93,40,95,38,89,36,91)(41,85,47,87,45,81,43,83)(42,84,48,86,46,88,44,82)(49,105,55,107,53,109,51,111)(50,112,56,106,54,108,52,110) );
G=PermutationGroup([[(1,61,18,49,39,41,10),(2,62,19,50,40,42,11),(3,63,20,51,33,43,12),(4,64,21,52,34,44,13),(5,57,22,53,35,45,14),(6,58,23,54,36,46,15),(7,59,24,55,37,47,16),(8,60,17,56,38,48,9),(25,79,108,91,88,71,100),(26,80,109,92,81,72,101),(27,73,110,93,82,65,102),(28,74,111,94,83,66,103),(29,75,112,95,84,67,104),(30,76,105,96,85,68,97),(31,77,106,89,86,69,98),(32,78,107,90,87,70,99)], [(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)], [(1,5),(3,7),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79),(82,86),(84,88),(89,93),(91,95),(98,102),(100,104),(106,110),(108,112)], [(1,97,7,99,5,101,3,103),(2,104,8,98,6,100,4,102),(9,69,15,71,13,65,11,67),(10,68,16,70,14,72,12,66),(17,77,23,79,21,73,19,75),(18,76,24,78,22,80,20,74),(25,64,27,62,29,60,31,58),(26,63,28,61,30,59,32,57),(33,94,39,96,37,90,35,92),(34,93,40,95,38,89,36,91),(41,85,47,87,45,81,43,83),(42,84,48,86,46,88,44,82),(49,105,55,107,53,109,51,111),(50,112,56,106,54,108,52,110)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | ··· | 8L | 14A | ··· | 14F | 14G | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28AD | 56A | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | - | ||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | Q8 | Q8 | C7×D4 | C7×Q8 | C7×Q8 | M4(2).C4 | C7×M4(2).C4 |
| kernel | C7×M4(2).C4 | C7×C8.C4 | C14×M4(2) | C7×M4(2) | M4(2).C4 | C8.C4 | C2×M4(2) | M4(2) | C2×C28 | C2×C28 | C22×C14 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 3 | 8 | 6 | 24 | 18 | 48 | 2 | 1 | 1 | 12 | 6 | 6 | 2 | 12 |
Matrix representation of C7×M4(2).C4 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 74 | 83 | 0 | 0 |
| 5 | 39 | 0 | 0 |
| 19 | 0 | 1 | 111 |
| 91 | 54 | 8 | 112 |
| 112 | 0 | 0 | 0 |
| 93 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 112 |
| 0 | 0 | 1 | 0 |
| 47 | 0 | 66 | 1 |
| 98 | 0 | 0 | 0 |
| 86 | 98 | 66 | 0 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[74,5,19,91,83,39,0,54,0,0,1,8,0,0,111,112],[112,93,0,0,0,1,0,0,0,0,1,1,0,0,0,112],[0,47,98,86,0,0,0,98,1,66,0,66,0,1,0,0] >;
C7×M4(2).C4 in GAP, Magma, Sage, TeX
C_7\times M_4(2).C_4
% in TeX
G:=Group("C7xM4(2).C4"); // GroupNames label
G:=SmallGroup(448,838);
// by ID
G=gap.SmallGroup(448,838);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,2403,9804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations