p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊19D4, C24.27C23, C23.440C24, C22.2292+ (1+4), C22.1762- (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, (C23×C4).393C22, (C22×C4).833C23, (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
Subgroups: 788 in 380 conjugacy classes, 120 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×34], C2×C4 [×18], C2×C4 [×32], D4 [×28], Q8 [×4], C23, C23 [×4], C23 [×26], C42 [×4], C42 [×2], C22⋊C4 [×28], C4⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×8], C22×C4 [×12], C2×D4 [×32], C2×Q8 [×6], C24 [×4], C2.C42 [×2], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4, C2×C4⋊C4 [×6], C4×D4 [×4], C4⋊D4 [×8], C22⋊Q8 [×8], C4.4D4 [×4], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C22×Q8, C4×C4⋊C4, C24.3C22 [×4], C23⋊2D4 [×2], C23.4Q8 [×2], C2×C4×D4, C2×C4⋊D4 [×2], C2×C22⋊Q8 [×2], C2×C4.4D4, C42⋊19D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C4⋊1D4 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ (1+4), 2- (1+4), C22.19C24, C2×C4⋊1D4, C22.31C24, D4⋊5D4 [×2], Q8⋊5D4 [×2], C42⋊19D4
Generators and relations
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 >
(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 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(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 6 51 63)(2 40 52 34)(3 8 49 61)(4 38 50 36)(5 22 62 12)(7 24 64 10)(9 39 23 33)(11 37 21 35)(13 17 27 31)(14 46 28 60)(15 19 25 29)(16 48 26 58)(18 56 32 42)(20 54 30 44)(41 45 55 59)(43 47 53 57)
(1 41)(2 16)(3 43)(4 14)(5 32)(6 59)(7 30)(8 57)(9 13)(10 44)(11 15)(12 42)(17 33)(18 62)(19 35)(20 64)(21 25)(22 56)(23 27)(24 54)(26 52)(28 50)(29 37)(31 39)(34 48)(36 46)(38 60)(40 58)(45 63)(47 61)(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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(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), (1,6,51,63)(2,40,52,34)(3,8,49,61)(4,38,50,36)(5,22,62,12)(7,24,64,10)(9,39,23,33)(11,37,21,35)(13,17,27,31)(14,46,28,60)(15,19,25,29)(16,48,26,58)(18,56,32,42)(20,54,30,44)(41,45,55,59)(43,47,53,57), (1,41)(2,16)(3,43)(4,14)(5,32)(6,59)(7,30)(8,57)(9,13)(10,44)(11,15)(12,42)(17,33)(18,62)(19,35)(20,64)(21,25)(22,56)(23,27)(24,54)(26,52)(28,50)(29,37)(31,39)(34,48)(36,46)(38,60)(40,58)(45,63)(47,61)(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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(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), (1,6,51,63)(2,40,52,34)(3,8,49,61)(4,38,50,36)(5,22,62,12)(7,24,64,10)(9,39,23,33)(11,37,21,35)(13,17,27,31)(14,46,28,60)(15,19,25,29)(16,48,26,58)(18,56,32,42)(20,54,30,44)(41,45,55,59)(43,47,53,57), (1,41)(2,16)(3,43)(4,14)(5,32)(6,59)(7,30)(8,57)(9,13)(10,44)(11,15)(12,42)(17,33)(18,62)(19,35)(20,64)(21,25)(22,56)(23,27)(24,54)(26,52)(28,50)(29,37)(31,39)(34,48)(36,46)(38,60)(40,58)(45,63)(47,61)(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)], [(1,13,9,41),(2,14,10,42),(3,15,11,43),(4,16,12,44),(5,48,38,20),(6,45,39,17),(7,46,40,18),(8,47,37,19),(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)], [(1,6,51,63),(2,40,52,34),(3,8,49,61),(4,38,50,36),(5,22,62,12),(7,24,64,10),(9,39,23,33),(11,37,21,35),(13,17,27,31),(14,46,28,60),(15,19,25,29),(16,48,26,58),(18,56,32,42),(20,54,30,44),(41,45,55,59),(43,47,53,57)], [(1,41),(2,16),(3,43),(4,14),(5,32),(6,59),(7,30),(8,57),(9,13),(10,44),(11,15),(12,42),(17,33),(18,62),(19,35),(20,64),(21,25),(22,56),(23,27),(24,54),(26,52),(28,50),(29,37),(31,39),(34,48),(36,46),(38,60),(40,58),(45,63),(47,61),(49,53),(51,55)])
Matrix representation ►G ⊆ 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] >;
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 |
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