direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×M4(2)⋊4C4, M4(2)⋊4C28, (C2×C8)⋊2C28, (C2×C56)⋊4C4, C28.54(C4⋊C4), (C2×C28).38Q8, (C2×C28).510D4, C22⋊C4.1C28, C22.3(C4×C28), C23.7(C2×C28), (C7×M4(2))⋊10C4, (C2×C14).10C42, C42⋊C2.3C14, (C2×M4(2)).9C14, C28.106(C22⋊C4), (C14×M4(2)).21C2, (C22×C28).389C22, C14.28(C2.C42), C4.5(C7×C4⋊C4), (C2×C4).3(C7×Q8), C22.6(C7×C4⋊C4), (C2×C4).15(C2×C28), (C2×C4).115(C7×D4), (C7×C22⋊C4).2C4, C4.27(C7×C22⋊C4), (C2×C14).23(C4⋊C4), (C2×C28).326(C2×C4), C22.9(C7×C22⋊C4), (C22×C4).19(C2×C14), (C22×C14).18(C2×C4), C2.9(C7×C2.C42), (C2×C14).72(C22⋊C4), (C7×C42⋊C2).17C2, SmallGroup(448,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×M4(2)⋊4C4
G = < a,b,c,d | a7=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=bc, dcd-1=b4c >
Subgroups: 138 in 90 conjugacy classes, 54 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C56, C2×C28, C2×C28, C22×C14, M4(2)⋊4C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C7×C42⋊C2, C14×M4(2), C7×M4(2)⋊4C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C2×C28, C7×D4, C7×Q8, M4(2)⋊4C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×M4(2)⋊4C4
►On 112 points
Generators in S112
(1 63 111 54 103 46 95)(2 64 112 55 104 47 96)(3 57 105 56 97 48 89)(4 58 106 49 98 41 90)(5 59 107 50 99 42 91)(6 60 108 51 100 43 92)(7 61 109 52 101 44 93)(8 62 110 53 102 45 94)(9 37 88 31 80 23 72)(10 38 81 32 73 24 65)(11 39 82 25 74 17 66)(12 40 83 26 75 18 67)(13 33 84 27 76 19 68)(14 34 85 28 77 20 69)(15 35 86 29 78 21 70)(16 36 87 30 79 22 71)
(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 65)(2 70)(3 67)(4 72)(5 69)(6 66)(7 71)(8 68)(9 58)(10 63)(11 60)(12 57)(13 62)(14 59)(15 64)(16 61)(17 92)(18 89)(19 94)(20 91)(21 96)(22 93)(23 90)(24 95)(25 100)(26 97)(27 102)(28 99)(29 104)(30 101)(31 98)(32 103)(33 110)(34 107)(35 112)(36 109)(37 106)(38 111)(39 108)(40 105)(41 80)(42 77)(43 74)(44 79)(45 76)(46 73)(47 78)(48 75)(49 88)(50 85)(51 82)(52 87)(53 84)(54 81)(55 86)(56 83)
(1 67 5 71)(2 4)(3 65 7 69)(6 8)(9 11)(10 61 14 57)(12 59 16 63)(13 15)(17 23)(18 91 22 95)(19 21)(20 89 24 93)(25 31)(26 99 30 103)(27 29)(28 97 32 101)(33 35)(34 105 38 109)(36 111 40 107)(37 39)(41 47)(42 79 46 75)(43 45)(44 77 48 73)(49 55)(50 87 54 83)(51 53)(52 85 56 81)(58 64)(60 62)(66 72)(68 70)(74 80)(76 78)(82 88)(84 86)(90 96)(92 94)(98 104)(100 102)(106 112)(108 110)
G:=sub<Sym(112)| (1,63,111,54,103,46,95)(2,64,112,55,104,47,96)(3,57,105,56,97,48,89)(4,58,106,49,98,41,90)(5,59,107,50,99,42,91)(6,60,108,51,100,43,92)(7,61,109,52,101,44,93)(8,62,110,53,102,45,94)(9,37,88,31,80,23,72)(10,38,81,32,73,24,65)(11,39,82,25,74,17,66)(12,40,83,26,75,18,67)(13,33,84,27,76,19,68)(14,34,85,28,77,20,69)(15,35,86,29,78,21,70)(16,36,87,30,79,22,71), (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), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,58)(10,63)(11,60)(12,57)(13,62)(14,59)(15,64)(16,61)(17,92)(18,89)(19,94)(20,91)(21,96)(22,93)(23,90)(24,95)(25,100)(26,97)(27,102)(28,99)(29,104)(30,101)(31,98)(32,103)(33,110)(34,107)(35,112)(36,109)(37,106)(38,111)(39,108)(40,105)(41,80)(42,77)(43,74)(44,79)(45,76)(46,73)(47,78)(48,75)(49,88)(50,85)(51,82)(52,87)(53,84)(54,81)(55,86)(56,83), (1,67,5,71)(2,4)(3,65,7,69)(6,8)(9,11)(10,61,14,57)(12,59,16,63)(13,15)(17,23)(18,91,22,95)(19,21)(20,89,24,93)(25,31)(26,99,30,103)(27,29)(28,97,32,101)(33,35)(34,105,38,109)(36,111,40,107)(37,39)(41,47)(42,79,46,75)(43,45)(44,77,48,73)(49,55)(50,87,54,83)(51,53)(52,85,56,81)(58,64)(60,62)(66,72)(68,70)(74,80)(76,78)(82,88)(84,86)(90,96)(92,94)(98,104)(100,102)(106,112)(108,110)>;
G:=Group( (1,63,111,54,103,46,95)(2,64,112,55,104,47,96)(3,57,105,56,97,48,89)(4,58,106,49,98,41,90)(5,59,107,50,99,42,91)(6,60,108,51,100,43,92)(7,61,109,52,101,44,93)(8,62,110,53,102,45,94)(9,37,88,31,80,23,72)(10,38,81,32,73,24,65)(11,39,82,25,74,17,66)(12,40,83,26,75,18,67)(13,33,84,27,76,19,68)(14,34,85,28,77,20,69)(15,35,86,29,78,21,70)(16,36,87,30,79,22,71), (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), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,58)(10,63)(11,60)(12,57)(13,62)(14,59)(15,64)(16,61)(17,92)(18,89)(19,94)(20,91)(21,96)(22,93)(23,90)(24,95)(25,100)(26,97)(27,102)(28,99)(29,104)(30,101)(31,98)(32,103)(33,110)(34,107)(35,112)(36,109)(37,106)(38,111)(39,108)(40,105)(41,80)(42,77)(43,74)(44,79)(45,76)(46,73)(47,78)(48,75)(49,88)(50,85)(51,82)(52,87)(53,84)(54,81)(55,86)(56,83), (1,67,5,71)(2,4)(3,65,7,69)(6,8)(9,11)(10,61,14,57)(12,59,16,63)(13,15)(17,23)(18,91,22,95)(19,21)(20,89,24,93)(25,31)(26,99,30,103)(27,29)(28,97,32,101)(33,35)(34,105,38,109)(36,111,40,107)(37,39)(41,47)(42,79,46,75)(43,45)(44,77,48,73)(49,55)(50,87,54,83)(51,53)(52,85,56,81)(58,64)(60,62)(66,72)(68,70)(74,80)(76,78)(82,88)(84,86)(90,96)(92,94)(98,104)(100,102)(106,112)(108,110) );
G=PermutationGroup([[(1,63,111,54,103,46,95),(2,64,112,55,104,47,96),(3,57,105,56,97,48,89),(4,58,106,49,98,41,90),(5,59,107,50,99,42,91),(6,60,108,51,100,43,92),(7,61,109,52,101,44,93),(8,62,110,53,102,45,94),(9,37,88,31,80,23,72),(10,38,81,32,73,24,65),(11,39,82,25,74,17,66),(12,40,83,26,75,18,67),(13,33,84,27,76,19,68),(14,34,85,28,77,20,69),(15,35,86,29,78,21,70),(16,36,87,30,79,22,71)], [(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)], [(1,65),(2,70),(3,67),(4,72),(5,69),(6,66),(7,71),(8,68),(9,58),(10,63),(11,60),(12,57),(13,62),(14,59),(15,64),(16,61),(17,92),(18,89),(19,94),(20,91),(21,96),(22,93),(23,90),(24,95),(25,100),(26,97),(27,102),(28,99),(29,104),(30,101),(31,98),(32,103),(33,110),(34,107),(35,112),(36,109),(37,106),(38,111),(39,108),(40,105),(41,80),(42,77),(43,74),(44,79),(45,76),(46,73),(47,78),(48,75),(49,88),(50,85),(51,82),(52,87),(53,84),(54,81),(55,86),(56,83)], [(1,67,5,71),(2,4),(3,65,7,69),(6,8),(9,11),(10,61,14,57),(12,59,16,63),(13,15),(17,23),(18,91,22,95),(19,21),(20,89,24,93),(25,31),(26,99,30,103),(27,29),(28,97,32,101),(33,35),(34,105,38,109),(36,111,40,107),(37,39),(41,47),(42,79,46,75),(43,45),(44,77,48,73),(49,55),(50,87,54,83),(51,53),(52,85,56,81),(58,64),(60,62),(66,72),(68,70),(74,80),(76,78),(82,88),(84,86),(90,96),(92,94),(98,104),(100,102),(106,112),(108,110)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14F | 14G | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28AD | 28AE | ··· | 28BB | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | C28 | D4 | Q8 | C7×D4 | C7×Q8 | M4(2)⋊4C4 | C7×M4(2)⋊4C4 |
| kernel | C7×M4(2)⋊4C4 | C7×C42⋊C2 | C14×M4(2) | C7×C22⋊C4 | C2×C56 | C7×M4(2) | M4(2)⋊4C4 | C42⋊C2 | C2×M4(2) | C22⋊C4 | C2×C8 | M4(2) | C2×C28 | C2×C28 | C2×C4 | C2×C4 | C7 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 24 | 24 | 24 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C7×M4(2)⋊4C4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 15 | 0 | 111 | 0 |
| 15 | 0 | 112 | 112 |
| 105 | 15 | 98 | 0 |
| 7 | 0 | 98 | 0 |
| 1 | 111 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 98 | 0 | 1 |
| 0 | 98 | 1 | 0 |
| 15 | 0 | 0 | 0 |
| 15 | 98 | 0 | 0 |
| 112 | 0 | 0 | 98 |
| 0 | 0 | 15 | 0 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,49,0,0,0,0,49],[15,15,105,7,0,0,15,0,111,112,98,98,0,112,0,0],[1,0,0,0,111,112,98,98,0,0,0,1,0,0,1,0],[15,15,112,0,0,98,0,0,0,0,0,15,0,0,98,0] >;
C7×M4(2)⋊4C4 in GAP, Magma, Sage, TeX
C_7\times M_4(2)\rtimes_4C_4
% in TeX
G:=Group("C7xM4(2):4C4"); // GroupNames label
G:=SmallGroup(448,148);
// by ID
G=gap.SmallGroup(448,148);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,7059,4911,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b*c,d*c*d^-1=b^4*c>;
// generators/relations