metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊7D7, (C4×D7)⋊2C4, C4.14(C4×D7), C28.11(C2×C4), D14⋊C4.4C2, C4⋊Dic7⋊12C2, D14.4(C2×C4), (C2×C4).44D14, C7⋊3(C42⋊C2), (C4×Dic7)⋊13C2, Dic7.9(C2×C4), C14.26(C4○D4), C2.4(D4⋊2D7), (C2×C14).33C23, C14.10(C22×C4), (C2×C28).56C22, C2.1(Q8⋊2D7), C22.17(C22×D7), (C2×Dic7).30C22, (C22×D7).19C22, (C7×C4⋊C4)⋊3C2, (C2×C4×D7).2C2, C2.12(C2×C4×D7), SmallGroup(224,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊7D7
G = < a,b,c,d | a4=b4=c7=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 278 in 76 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C42⋊C2, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C4⋊C4⋊7D7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, D14, C42⋊C2, C4×D7, C22×D7, C2×C4×D7, D4⋊2D7, Q8⋊2D7, C4⋊C4⋊7D7
►On 112 points
(1 83 13 76)(2 84 14 77)(3 78 8 71)(4 79 9 72)(5 80 10 73)(6 81 11 74)(7 82 12 75)(15 64 22 57)(16 65 23 58)(17 66 24 59)(18 67 25 60)(19 68 26 61)(20 69 27 62)(21 70 28 63)(29 106 36 99)(30 107 37 100)(31 108 38 101)(32 109 39 102)(33 110 40 103)(34 111 41 104)(35 112 42 105)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)
(1 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 106 71 92)(58 107 72 93)(59 108 73 94)(60 109 74 95)(61 110 75 96)(62 111 76 97)(63 112 77 98)(64 99 78 85)(65 100 79 86)(66 101 80 87)(67 102 81 88)(68 103 82 89)(69 104 83 90)(70 105 84 91)
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)
G:=sub<Sym(112)| (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)>;
G:=Group( (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109) );
G=PermutationGroup([[(1,83,13,76),(2,84,14,77),(3,78,8,71),(4,79,9,72),(5,80,10,73),(6,81,11,74),(7,82,12,75),(15,64,22,57),(16,65,23,58),(17,66,24,59),(18,67,25,60),(19,68,26,61),(20,69,27,62),(21,70,28,63),(29,106,36,99),(30,107,37,100),(31,108,38,101),(32,109,39,102),(33,110,40,103),(34,111,41,104),(35,112,42,105),(43,92,50,85),(44,93,51,86),(45,94,52,87),(46,95,53,88),(47,96,54,89),(48,97,55,90),(49,98,56,91)], [(1,48,20,34),(2,49,21,35),(3,43,15,29),(4,44,16,30),(5,45,17,31),(6,46,18,32),(7,47,19,33),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,106,71,92),(58,107,72,93),(59,108,73,94),(60,109,74,95),(61,110,75,96),(62,111,76,97),(63,112,77,98),(64,99,78,85),(65,100,79,86),(66,101,80,87),(67,102,81,88),(68,103,82,89),(69,104,83,90),(70,105,84,91)], [(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,59),(60,63),(61,62),(64,66),(67,70),(68,69),(71,73),(74,77),(75,76),(78,80),(81,84),(82,83),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109)]])
C4⋊C4⋊7D7 is a maximal subgroup of
(D4×D7)⋊C4 D4⋊2D7⋊C4 C8⋊Dic7⋊C2 C56⋊1C4⋊C2 (Q8×D7)⋊C4 Q8⋊2D7⋊C4 D14⋊C8.C2 (C2×C8).D14 (C8×D7)⋊C4 C8⋊(C4×D7) C4.Q8⋊D7 C28.(C4○D4) C8.27(C4×D7) C56⋊(C2×C4) C2.D8⋊D7 C2.D8⋊7D7 C14.82+ 1+4 C14.52- 1+4 C14.112+ 1+4 D7×C42⋊C2 C42.91D14 C42.97D14 C42.98D14 C4×D4⋊2D7 C42⋊11D14 C42.113D14 C42.117D14 C42.125D14 C4×Q8⋊2D7 C42.132D14 C42.135D14 C4⋊C4⋊21D14 C14.382+ 1+4 C14.452+ 1+4 C14.1152+ 1+4 C22⋊Q8⋊25D7 C4⋊C4⋊26D14 C14.162- 1+4 C14.532+ 1+4 C14.212- 1+4 C14.222- 1+4 C14.232- 1+4 C14.772- 1+4 C4⋊C4.197D14 C14.1222+ 1+4 C14.622+ 1+4 C14.662+ 1+4 C42.236D14 C42.148D14 C42.237D14 C42.151D14 C42.152D14 C42.153D14 C42.154D14 C42⋊24D14 C42.189D14 C42.162D14 C42.164D14 C42.241D14 C42.174D14 C42.177D14 C42.179D14
C4⋊C4⋊7D7 is a maximal quotient of
Dic7.5C42 C7⋊(C42⋊5C4) C14.(C4×D4) C22.58(D4×D7) D14⋊C42 C2.(C4×D28) C42.200D14 C42.202D14 C42.31D14 C4⋊C4×Dic7 (C4×Dic7)⋊9C4 C22.23(Q8×D7) C4⋊(D14⋊C4) D14⋊C4⋊7C4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 2 | ··· | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | D14 | C4×D7 | D4⋊2D7 | Q8⋊2D7 |
| kernel | C4⋊C4⋊7D7 | C4×Dic7 | C4⋊Dic7 | D14⋊C4 | C7×C4⋊C4 | C2×C4×D7 | C4×D7 | C4⋊C4 | C14 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 3 | 4 | 9 | 12 | 3 | 3 |
Matrix representation of C4⋊C4⋊7D7 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 1 | 17 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 17 |
| 0 | 1 | 0 | 0 |
| 28 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 17 | 28 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,12,1,0,0,0,17],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,2,17],[0,28,0,0,1,7,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,17,0,0,0,28] >;
C4⋊C4⋊7D7 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_7D_7
% in TeX
G:=Group("C4:C4:7D7"); // GroupNames label
G:=SmallGroup(224,87);
// by ID
G=gap.SmallGroup(224,87);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,362,188,50,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^7=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations