metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊3Dic7, (C4×C28)⋊3C4, (D4×C14)⋊2C4, (C2×D4)⋊2Dic7, C4⋊1D4.2D7, C7⋊2(C42⋊C4), (C2×D4).10D14, C23⋊Dic7⋊8C2, (C22×C14).17D4, C23.8(C7⋊D4), C14.25(C23⋊C4), (D4×C14).173C22, C2.10(C23⋊Dic7), C22.16(C23.D7), (C2×C28).10(C2×C4), (C7×C4⋊1D4).7C2, (C2×C4).3(C2×Dic7), (C2×C14).103(C22⋊C4), SmallGroup(448,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊3Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 428 in 86 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C14, C14, C42, C22⋊C4, C2×D4, C2×D4, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C4⋊1D4, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C42⋊C4, C23.D7, C4×C28, D4×C14, D4×C14, C23⋊Dic7, C7×C4⋊1D4, C42⋊3Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C42⋊C4, C23.D7, C23⋊Dic7, C42⋊3Dic7
►On 56 points
(1 54)(2 55)(3 56)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 31 38 22)(16 23 39 32)(17 33 40 24)(18 25 41 34)(19 35 42 26)(20 27 29 36)(21 37 30 28)
(1 8 54 47)(2 48 55 9)(3 10 56 49)(4 50 43 11)(5 12 44 51)(6 52 45 13)(7 14 46 53)(15 22 38 31)(16 32 39 23)(17 24 40 33)(18 34 41 25)(19 26 42 35)(20 36 29 27)(21 28 30 37)
(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)
(1 30 8 37)(2 29 9 36)(3 42 10 35)(4 41 11 34)(5 40 12 33)(6 39 13 32)(7 38 14 31)(15 53 22 46)(16 52 23 45)(17 51 24 44)(18 50 25 43)(19 49 26 56)(20 48 27 55)(21 47 28 54)
G:=sub<Sym(56)| (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,31,38,22)(16,23,39,32)(17,33,40,24)(18,25,41,34)(19,35,42,26)(20,27,29,36)(21,37,30,28), (1,8,54,47)(2,48,55,9)(3,10,56,49)(4,50,43,11)(5,12,44,51)(6,52,45,13)(7,14,46,53)(15,22,38,31)(16,32,39,23)(17,24,40,33)(18,34,41,25)(19,26,42,35)(20,36,29,27)(21,28,30,37), (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), (1,30,8,37)(2,29,9,36)(3,42,10,35)(4,41,11,34)(5,40,12,33)(6,39,13,32)(7,38,14,31)(15,53,22,46)(16,52,23,45)(17,51,24,44)(18,50,25,43)(19,49,26,56)(20,48,27,55)(21,47,28,54)>;
G:=Group( (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,31,38,22)(16,23,39,32)(17,33,40,24)(18,25,41,34)(19,35,42,26)(20,27,29,36)(21,37,30,28), (1,8,54,47)(2,48,55,9)(3,10,56,49)(4,50,43,11)(5,12,44,51)(6,52,45,13)(7,14,46,53)(15,22,38,31)(16,32,39,23)(17,24,40,33)(18,34,41,25)(19,26,42,35)(20,36,29,27)(21,28,30,37), (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), (1,30,8,37)(2,29,9,36)(3,42,10,35)(4,41,11,34)(5,40,12,33)(6,39,13,32)(7,38,14,31)(15,53,22,46)(16,52,23,45)(17,51,24,44)(18,50,25,43)(19,49,26,56)(20,48,27,55)(21,47,28,54) );
G=PermutationGroup([[(1,54),(2,55),(3,56),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,31,38,22),(16,23,39,32),(17,33,40,24),(18,25,41,34),(19,35,42,26),(20,27,29,36),(21,37,30,28)], [(1,8,54,47),(2,48,55,9),(3,10,56,49),(4,50,43,11),(5,12,44,51),(6,52,45,13),(7,14,46,53),(15,22,38,31),(16,32,39,23),(17,24,40,33),(18,34,41,25),(19,26,42,35),(20,36,29,27),(21,28,30,37)], [(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)], [(1,30,8,37),(2,29,9,36),(3,42,10,35),(4,41,11,34),(5,40,12,33),(6,39,13,32),(7,38,14,31),(15,53,22,46),(16,52,23,45),(17,51,24,44),(18,50,25,43),(19,49,26,56),(20,48,27,55),(21,47,28,54)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D7 | Dic7 | Dic7 | D14 | C7⋊D4 | C23⋊C4 | C42⋊C4 | C23⋊Dic7 | C42⋊3Dic7 |
| kernel | C42⋊3Dic7 | C23⋊Dic7 | C7×C4⋊1D4 | C4×C28 | D4×C14 | C22×C14 | C4⋊1D4 | C42 | C2×D4 | C2×D4 | C23 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of C42⋊3Dic7 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 20 | 3 |
| 0 | 0 | 21 | 9 |
| 20 | 3 | 0 | 0 |
| 21 | 9 | 0 | 0 |
| 0 | 0 | 9 | 26 |
| 0 | 0 | 8 | 20 |
| 13 | 15 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 4 | 18 |
| 0 | 0 | 14 | 25 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 26 | 0 | 0 |
| 17 | 20 | 0 | 0 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,20,21,0,0,3,9],[20,21,0,0,3,9,0,0,0,0,9,8,0,0,26,20],[13,2,0,0,15,16,0,0,0,0,4,14,0,0,18,25],[0,0,9,17,0,0,26,20,1,0,0,0,0,1,0,0] >;
C42⋊3Dic7 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3{\rm Dic}_7 % in TeX
G:=Group("C4^2:3Dic7"); // GroupNames label
G:=SmallGroup(448,102);
// by ID
G=gap.SmallGroup(448,102);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,1571,570,297,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations