p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊18D4, C24.320C23, C23.437C24, C22.2262+ (1+4), C22.1742- (1+4), (C2×D4)⋊34D4, C2.28(D42), C4⋊5(C4⋊D4), C23.47(C2×D4), C42⋊9C4⋊26C2, C2.49(D4⋊6D4), (C22×C4).94C23, C23.7Q8⋊65C2, C23.10D4⋊39C2, (C23×C4).390C22, (C2×C42).543C22, C22.288(C22×D4), (C22×D4).161C22, C2.C42.180C22, C2.24(C22.49C24), C2.12(C22.31C24), (C2×C4×D4)⋊43C2, (C2×C4⋊D4)⋊15C2, (C2×C4).351(C2×D4), C2.32(C2×C4⋊D4), (C2×C4).819(C4○D4), (C2×C4⋊C4).297C22, C22.314(C2×C4○D4), (C2×C22⋊C4).173C22, SmallGroup(128,1269)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 804 in 394 conjugacy classes, 124 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×10], C22 [×3], C22 [×4], C22 [×40], C2×C4 [×14], C2×C4 [×42], D4 [×32], C23, C23 [×8], C23 [×24], C42 [×4], C22⋊C4 [×24], C4⋊C4 [×14], C22×C4, C22×C4 [×10], C22×C4 [×24], C2×D4 [×8], C2×D4 [×28], C24 [×4], C2.C42 [×4], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×8], C4×D4 [×8], C4⋊D4 [×16], C23×C4 [×4], C22×D4 [×6], C23.7Q8 [×4], C42⋊9C4, C23.10D4 [×4], C2×C4×D4 [×2], C2×C4⋊D4 [×4], C42⋊18D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C4⋊D4 [×8], C22×D4 [×3], C2×C4○D4 [×2], 2+ (1+4), 2- (1+4), C2×C4⋊D4 [×2], C22.31C24, D42, D4⋊6D4 [×2], C22.49C24, C42⋊18D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, 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 15 11 20)(2 16 12 17)(3 13 9 18)(4 14 10 19)(5 29 28 63)(6 30 25 64)(7 31 26 61)(8 32 27 62)(21 44 45 52)(22 41 46 49)(23 42 47 50)(24 43 48 51)(33 38 55 59)(34 39 56 60)(35 40 53 57)(36 37 54 58)
(1 32 44 39)(2 31 41 38)(3 30 42 37)(4 29 43 40)(5 48 35 14)(6 47 36 13)(7 46 33 16)(8 45 34 15)(9 64 50 58)(10 63 51 57)(11 62 52 60)(12 61 49 59)(17 26 22 55)(18 25 23 54)(19 28 24 53)(20 27 21 56)
(1 33)(2 56)(3 35)(4 54)(5 42)(6 51)(7 44)(8 49)(9 53)(10 36)(11 55)(12 34)(13 57)(14 37)(15 59)(16 39)(17 60)(18 40)(19 58)(20 38)(21 31)(22 62)(23 29)(24 64)(25 43)(26 52)(27 41)(28 50)(30 48)(32 46)(45 61)(47 63)
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), (1,15,11,20)(2,16,12,17)(3,13,9,18)(4,14,10,19)(5,29,28,63)(6,30,25,64)(7,31,26,61)(8,32,27,62)(21,44,45,52)(22,41,46,49)(23,42,47,50)(24,43,48,51)(33,38,55,59)(34,39,56,60)(35,40,53,57)(36,37,54,58), (1,32,44,39)(2,31,41,38)(3,30,42,37)(4,29,43,40)(5,48,35,14)(6,47,36,13)(7,46,33,16)(8,45,34,15)(9,64,50,58)(10,63,51,57)(11,62,52,60)(12,61,49,59)(17,26,22,55)(18,25,23,54)(19,28,24,53)(20,27,21,56), (1,33)(2,56)(3,35)(4,54)(5,42)(6,51)(7,44)(8,49)(9,53)(10,36)(11,55)(12,34)(13,57)(14,37)(15,59)(16,39)(17,60)(18,40)(19,58)(20,38)(21,31)(22,62)(23,29)(24,64)(25,43)(26,52)(27,41)(28,50)(30,48)(32,46)(45,61)(47,63)>;
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), (1,15,11,20)(2,16,12,17)(3,13,9,18)(4,14,10,19)(5,29,28,63)(6,30,25,64)(7,31,26,61)(8,32,27,62)(21,44,45,52)(22,41,46,49)(23,42,47,50)(24,43,48,51)(33,38,55,59)(34,39,56,60)(35,40,53,57)(36,37,54,58), (1,32,44,39)(2,31,41,38)(3,30,42,37)(4,29,43,40)(5,48,35,14)(6,47,36,13)(7,46,33,16)(8,45,34,15)(9,64,50,58)(10,63,51,57)(11,62,52,60)(12,61,49,59)(17,26,22,55)(18,25,23,54)(19,28,24,53)(20,27,21,56), (1,33)(2,56)(3,35)(4,54)(5,42)(6,51)(7,44)(8,49)(9,53)(10,36)(11,55)(12,34)(13,57)(14,37)(15,59)(16,39)(17,60)(18,40)(19,58)(20,38)(21,31)(22,62)(23,29)(24,64)(25,43)(26,52)(27,41)(28,50)(30,48)(32,46)(45,61)(47,63) );
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)], [(1,15,11,20),(2,16,12,17),(3,13,9,18),(4,14,10,19),(5,29,28,63),(6,30,25,64),(7,31,26,61),(8,32,27,62),(21,44,45,52),(22,41,46,49),(23,42,47,50),(24,43,48,51),(33,38,55,59),(34,39,56,60),(35,40,53,57),(36,37,54,58)], [(1,32,44,39),(2,31,41,38),(3,30,42,37),(4,29,43,40),(5,48,35,14),(6,47,36,13),(7,46,33,16),(8,45,34,15),(9,64,50,58),(10,63,51,57),(11,62,52,60),(12,61,49,59),(17,26,22,55),(18,25,23,54),(19,28,24,53),(20,27,21,56)], [(1,33),(2,56),(3,35),(4,54),(5,42),(6,51),(7,44),(8,49),(9,53),(10,36),(11,55),(12,34),(13,57),(14,37),(15,59),(16,39),(17,60),(18,40),(19,58),(20,38),(21,31),(22,62),(23,29),(24,64),(25,43),(26,52),(27,41),(28,50),(30,48),(32,46),(45,61),(47,63)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42⋊18D4 | C23.7Q8 | C42⋊9C4 | C23.10D4 | C2×C4×D4 | C2×C4⋊D4 | C42 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 4 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{18}D_4 % in TeX
G:=Group("C4^2:18D4"); // GroupNames label
G:=SmallGroup(128,1269);
// by ID
G=gap.SmallGroup(128,1269);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675,80]);
// 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^-1,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations