p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊19D4, C24.27C23, C23.440C24, C22.1762- 1+4, C22.2292+ 1+4, C4⋊C4⋊25D4, C4.36(C4⋊1D4), C23⋊2D4⋊18C2, C2.68(D4⋊5D4), C2.37(Q8⋊5D4), C23.51(C4○D4), C23.4Q8⋊20C2, (C22×C4).833C23, (C23×C4).393C22, (C2×C42).545C22, C22.291(C22×D4), C24.3C22⋊53C2, (C22×D4).162C22, (C22×Q8).128C22, C2.61(C22.19C24), C2.C42.546C22, C2.14(C22.31C24), (C2×C4×D4)⋊44C2, (C4×C4⋊C4)⋊85C2, (C2×C4).71(C2×D4), (C2×C4⋊D4)⋊17C2, C2.10(C2×C4⋊1D4), (C2×C22⋊Q8)⋊21C2, (C2×C4.4D4)⋊14C2, (C2×C4⋊C4).299C22, C22.317(C2×C4○D4), (C2×C22⋊C4).175C22, SmallGroup(128,1272)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊19D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 788 in 380 conjugacy classes, 120 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C4.4D4, C23×C4, C22×D4, C22×D4, C22×Q8, C4×C4⋊C4, C24.3C22, C23⋊2D4, C23.4Q8, C2×C4×D4, C2×C4⋊D4, C2×C22⋊Q8, C2×C4.4D4, C42⋊19D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊1D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C2×C4⋊1D4, C22.31C24, D4⋊5D4, Q8⋊5D4, C42⋊19D4
►On 64 points
(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)
(1 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 18 38 46)(6 19 39 47)(7 20 40 48)(8 17 37 45)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(1 39 51 63)(2 7 52 34)(3 37 49 61)(4 5 50 36)(6 23 33 11)(8 21 35 9)(10 38 22 62)(12 40 24 64)(13 17 25 29)(14 46 26 58)(15 19 27 31)(16 48 28 60)(18 54 30 42)(20 56 32 44)(41 45 53 57)(43 47 55 59)
(1 43)(2 14)(3 41)(4 16)(5 60)(6 31)(7 58)(8 29)(9 13)(10 44)(11 15)(12 42)(17 35)(18 64)(19 33)(20 62)(21 25)(22 56)(23 27)(24 54)(26 52)(28 50)(30 40)(32 38)(34 46)(36 48)(37 57)(39 59)(45 61)(47 63)(49 53)(51 55)
G:=sub<Sym(64)| (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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,39,51,63)(2,7,52,34)(3,37,49,61)(4,5,50,36)(6,23,33,11)(8,21,35,9)(10,38,22,62)(12,40,24,64)(13,17,25,29)(14,46,26,58)(15,19,27,31)(16,48,28,60)(18,54,30,42)(20,56,32,44)(41,45,53,57)(43,47,55,59), (1,43)(2,14)(3,41)(4,16)(5,60)(6,31)(7,58)(8,29)(9,13)(10,44)(11,15)(12,42)(17,35)(18,64)(19,33)(20,62)(21,25)(22,56)(23,27)(24,54)(26,52)(28,50)(30,40)(32,38)(34,46)(36,48)(37,57)(39,59)(45,61)(47,63)(49,53)(51,55)>;
G:=Group( (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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,39,51,63)(2,7,52,34)(3,37,49,61)(4,5,50,36)(6,23,33,11)(8,21,35,9)(10,38,22,62)(12,40,24,64)(13,17,25,29)(14,46,26,58)(15,19,27,31)(16,48,28,60)(18,54,30,42)(20,56,32,44)(41,45,53,57)(43,47,55,59), (1,43)(2,14)(3,41)(4,16)(5,60)(6,31)(7,58)(8,29)(9,13)(10,44)(11,15)(12,42)(17,35)(18,64)(19,33)(20,62)(21,25)(22,56)(23,27)(24,54)(26,52)(28,50)(30,40)(32,38)(34,46)(36,48)(37,57)(39,59)(45,61)(47,63)(49,53)(51,55) );
G=PermutationGroup([[(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)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,18,38,46),(6,19,39,47),(7,20,40,48),(8,17,37,45),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,35),(30,62,58,36),(31,63,59,33),(32,64,60,34)], [(1,39,51,63),(2,7,52,34),(3,37,49,61),(4,5,50,36),(6,23,33,11),(8,21,35,9),(10,38,22,62),(12,40,24,64),(13,17,25,29),(14,46,26,58),(15,19,27,31),(16,48,28,60),(18,54,30,42),(20,56,32,44),(41,45,53,57),(43,47,55,59)], [(1,43),(2,14),(3,41),(4,16),(5,60),(6,31),(7,58),(8,29),(9,13),(10,44),(11,15),(12,42),(17,35),(18,64),(19,33),(20,62),(21,25),(22,56),(23,27),(24,54),(26,52),(28,50),(30,40),(32,38),(34,46),(36,48),(37,57),(39,59),(45,61),(47,63),(49,53),(51,55)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊19D4 | C4×C4⋊C4 | C24.3C22 | C23⋊2D4 | C23.4Q8 | C2×C4×D4 | C2×C4⋊D4 | C2×C22⋊Q8 | C2×C4.4D4 | C42 | C4⋊C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 8 | 8 | 1 | 1 |
Matrix representation of C42⋊19D4 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,3,0,0,0,0,4,3,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,2,3,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,1,4,0,0,0,0,0,4] >;
C42⋊19D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{19}D_4 % in TeX
G:=Group("C4^2:19D4"); // GroupNames label
G:=SmallGroup(128,1272);
// by ID
G=gap.SmallGroup(128,1272);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations