metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊6D7, C4⋊Dic7⋊7C2, D14⋊C4.5C2, (C2×C4).32D14, Dic7⋊C4⋊13C2, C7⋊3(C42⋊2C2), (C4×Dic7)⋊14C2, (C2×C28).7C22, C2.16(C4○D28), C14.14(C4○D4), (C2×C14).39C23, C2.7(Q8⋊2D7), C2.14(D4⋊2D7), (C22×D7).8C22, C22.53(C22×D7), (C2×Dic7).32C22, (C7×C4⋊C4)⋊9C2, SmallGroup(224,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊D7
G = < a,b,c,d | a4=b4=c7=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 246 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, Dic7, C28, D14, C2×C14, C42⋊2C2, C2×Dic7, C2×C28, C22×D7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C4⋊C4⋊D7
Quotients: C1, C2, C22, C23, D7, C4○D4, D14, C42⋊2C2, C22×D7, C4○D28, D4⋊2D7, Q8⋊2D7, C4⋊C4⋊D7
►On 112 points
(1 55 13 48)(2 56 14 49)(3 50 8 43)(4 51 9 44)(5 52 10 45)(6 53 11 46)(7 54 12 47)(15 36 22 29)(16 37 23 30)(17 38 24 31)(18 39 25 32)(19 40 26 33)(20 41 27 34)(21 42 28 35)(57 99 64 106)(58 100 65 107)(59 101 66 108)(60 102 67 109)(61 103 68 110)(62 104 69 111)(63 105 70 112)(71 85 78 92)(72 86 79 93)(73 87 80 94)(74 88 81 95)(75 89 82 96)(76 90 83 97)(77 91 84 98)
(1 76 20 62)(2 77 21 63)(3 71 15 57)(4 72 16 58)(5 73 17 59)(6 74 18 60)(7 75 19 61)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)
(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 45)(30 44)(31 43)(32 49)(33 48)(34 47)(35 46)(36 52)(37 51)(38 50)(39 56)(40 55)(41 54)(42 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 108)(86 107)(87 106)(88 112)(89 111)(90 110)(91 109)(92 101)(93 100)(94 99)(95 105)(96 104)(97 103)(98 102)
G:=sub<Sym(112)| (1,55,13,48)(2,56,14,49)(3,50,8,43)(4,51,9,44)(5,52,10,45)(6,53,11,46)(7,54,12,47)(15,36,22,29)(16,37,23,30)(17,38,24,31)(18,39,25,32)(19,40,26,33)(20,41,27,34)(21,42,28,35)(57,99,64,106)(58,100,65,107)(59,101,66,108)(60,102,67,109)(61,103,68,110)(62,104,69,111)(63,105,70,112)(71,85,78,92)(72,86,79,93)(73,87,80,94)(74,88,81,95)(75,89,82,96)(76,90,83,97)(77,91,84,98), (1,76,20,62)(2,77,21,63)(3,71,15,57)(4,72,16,58)(5,73,17,59)(6,74,18,60)(7,75,19,61)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98), (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)(29,45)(30,44)(31,43)(32,49)(33,48)(34,47)(35,46)(36,52)(37,51)(38,50)(39,56)(40,55)(41,54)(42,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,108)(86,107)(87,106)(88,112)(89,111)(90,110)(91,109)(92,101)(93,100)(94,99)(95,105)(96,104)(97,103)(98,102)>;
G:=Group( (1,55,13,48)(2,56,14,49)(3,50,8,43)(4,51,9,44)(5,52,10,45)(6,53,11,46)(7,54,12,47)(15,36,22,29)(16,37,23,30)(17,38,24,31)(18,39,25,32)(19,40,26,33)(20,41,27,34)(21,42,28,35)(57,99,64,106)(58,100,65,107)(59,101,66,108)(60,102,67,109)(61,103,68,110)(62,104,69,111)(63,105,70,112)(71,85,78,92)(72,86,79,93)(73,87,80,94)(74,88,81,95)(75,89,82,96)(76,90,83,97)(77,91,84,98), (1,76,20,62)(2,77,21,63)(3,71,15,57)(4,72,16,58)(5,73,17,59)(6,74,18,60)(7,75,19,61)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98), (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)(29,45)(30,44)(31,43)(32,49)(33,48)(34,47)(35,46)(36,52)(37,51)(38,50)(39,56)(40,55)(41,54)(42,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,108)(86,107)(87,106)(88,112)(89,111)(90,110)(91,109)(92,101)(93,100)(94,99)(95,105)(96,104)(97,103)(98,102) );
G=PermutationGroup([[(1,55,13,48),(2,56,14,49),(3,50,8,43),(4,51,9,44),(5,52,10,45),(6,53,11,46),(7,54,12,47),(15,36,22,29),(16,37,23,30),(17,38,24,31),(18,39,25,32),(19,40,26,33),(20,41,27,34),(21,42,28,35),(57,99,64,106),(58,100,65,107),(59,101,66,108),(60,102,67,109),(61,103,68,110),(62,104,69,111),(63,105,70,112),(71,85,78,92),(72,86,79,93),(73,87,80,94),(74,88,81,95),(75,89,82,96),(76,90,83,97),(77,91,84,98)], [(1,76,20,62),(2,77,21,63),(3,71,15,57),(4,72,16,58),(5,73,17,59),(6,74,18,60),(7,75,19,61),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,99,43,85),(30,100,44,86),(31,101,45,87),(32,102,46,88),(33,103,47,89),(34,104,48,90),(35,105,49,91),(36,106,50,92),(37,107,51,93),(38,108,52,94),(39,109,53,95),(40,110,54,96),(41,111,55,97),(42,112,56,98)], [(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),(29,45),(30,44),(31,43),(32,49),(33,48),(34,47),(35,46),(36,52),(37,51),(38,50),(39,56),(40,55),(41,54),(42,53),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,108),(86,107),(87,106),(88,112),(89,111),(90,110),(91,109),(92,101),(93,100),(94,99),(95,105),(96,104),(97,103),(98,102)]])
C4⋊C4⋊D7 is a maximal subgroup of
C14.52- 1+4 C14.112+ 1+4 C14.62- 1+4 C42⋊10D14 C42.93D14 C42.96D14 C42.99D14 C42.102D14 C42⋊16D14 C42⋊17D14 C42.119D14 C42.122D14 C42.131D14 C42.134D14 C42.135D14 C14.342+ 1+4 C14.422+ 1+4 C14.462+ 1+4 C14.482+ 1+4 C22⋊Q8⋊25D7 C14.532+ 1+4 C14.202- 1+4 C14.222- 1+4 C14.232- 1+4 C14.772- 1+4 C14.242- 1+4 C14.562+ 1+4 C14.572+ 1+4 C14.582+ 1+4 C4⋊C4.197D14 C14.612+ 1+4 C14.1222+ 1+4 C14.642+ 1+4 C14.842- 1+4 C14.852- 1+4 C14.682+ 1+4 C42.237D14 C42.150D14 C42.151D14 C42.152D14 C42.153D14 C42.154D14 C42.155D14 C42.157D14 C42.160D14 D7×C42⋊2C2 C42⋊23D14 C42.161D14 C42.163D14 C42.164D14 C42.165D14 C42.176D14 C42.177D14 C42.178D14 C42.180D14
C4⋊C4⋊D7 is a maximal quotient of
C7⋊(C42⋊5C4) C4⋊Dic7⋊8C4 (C2×C4).Dic14 (C22×C4).D14 D14⋊C4⋊5C4 C2.(C4×D28) (C2×C4).21D28 (C22×D7).9D4 C22.23(Q8×D7) C4⋊C4⋊5Dic7 (C2×C28).288D4 (C2×C28).55D4 D14⋊C4⋊7C4 (C2×C28).290D4 (C2×C4).45D28
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | C4○D28 | D4⋊2D7 | Q8⋊2D7 |
| kernel | C4⋊C4⋊D7 | C4×Dic7 | Dic7⋊C4 | C4⋊Dic7 | D14⋊C4 | C7×C4⋊C4 | C4⋊C4 | C14 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 3 | 6 | 9 | 12 | 3 | 3 |
Matrix representation of C4⋊C4⋊D7 ►in GL4(𝔽29) generated by
| 20 | 14 | 0 | 0 |
| 15 | 9 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 1 |
| 7 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 |
G:=sub<GL(4,GF(29))| [20,15,0,0,14,9,0,0,0,0,17,12,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,28,0,0,0,27,1],[7,28,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,28,0,0,0,28] >;
C4⋊C4⋊D7 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes D_7
% in TeX
G:=Group("C4:C4:D7"); // GroupNames label
G:=SmallGroup(224,93);
// by ID
G=gap.SmallGroup(224,93);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,55,218,188,86,6917]);
// 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,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations