p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.294D4, C4.22- (1+4), C42.428C23, C4.212+ (1+4), (C2×C4)⋊8SD16, C8⋊8D4⋊29C2, Q8⋊Q8⋊39C2, D4⋊2Q8⋊37C2, C4⋊SD16⋊38C2, C4.88(C2×SD16), D4.D4⋊39C2, C4⋊C4.185C23, C4⋊C8.339C22, (C2×C8).338C23, (C2×C4).444C24, C23.403(C2×D4), (C22×C4).522D4, C4⋊Q8.323C22, C22.4(C2×SD16), C4.Q8.91C22, (C4×D4).125C22, (C2×D4).187C23, (C4×Q8).121C22, (C2×Q8).174C23, C2.26(C22×SD16), C4⋊D4.207C22, C4⋊1D4.175C22, (C22×C8).351C22, (C2×C42).901C22, (C2×SD16).87C22, C22.704(C22×D4), C22⋊Q8.211C22, D4⋊C4.113C22, C2.69(D8⋊C22), (C22×C4).1577C23, Q8⋊C4.108C22, C23.37C23⋊24C2, C22.26C24.48C2, C2.63(C22.31C24), (C2×C4⋊C8)⋊38C2, (C2×C4).568(C2×D4), SmallGroup(128,1978)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 196 conjugacy classes, 94 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×13], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×2], SD16 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C4.Q8 [×4], C2×C42, C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8 [×3], C22×C8 [×2], C2×SD16 [×4], C2×C4○D4, C2×C4⋊C8, C4⋊SD16 [×2], D4.D4 [×2], C8⋊8D4 [×4], Q8⋊Q8 [×2], D4⋊2Q8 [×2], C22.26C24, C23.37C23, C42.294D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, C22×SD16, D8⋊C22, C42.294D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c3 >
►On 64 points
(1 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 40 13 36)(10 33 14 37)(11 34 15 38)(12 35 16 39)(17 32 21 28)(18 25 22 29)(19 26 23 30)(20 27 24 31)(41 56 45 52)(42 49 46 53)(43 50 47 54)(44 51 48 55)
(1 32 55 15)(2 12 56 29)(3 26 49 9)(4 14 50 31)(5 28 51 11)(6 16 52 25)(7 30 53 13)(8 10 54 27)(17 48 34 57)(18 62 35 45)(19 42 36 59)(20 64 37 47)(21 44 38 61)(22 58 39 41)(23 46 40 63)(24 60 33 43)
(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 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)(17 33 21 37)(18 36 22 40)(19 39 23 35)(20 34 24 38)(25 26 29 30)(27 32 31 28)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)
G:=sub<Sym(64)| (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31)(41,56,45,52)(42,49,46,53)(43,50,47,54)(44,51,48,55), (1,32,55,15)(2,12,56,29)(3,26,49,9)(4,14,50,31)(5,28,51,11)(6,16,52,25)(7,30,53,13)(8,10,54,27)(17,48,34,57)(18,62,35,45)(19,42,36,59)(20,64,37,47)(21,44,38,61)(22,58,39,41)(23,46,40,63)(24,60,33,43), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,33,21,37)(18,36,22,40)(19,39,23,35)(20,34,24,38)(25,26,29,30)(27,32,31,28)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55)>;
G:=Group( (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31)(41,56,45,52)(42,49,46,53)(43,50,47,54)(44,51,48,55), (1,32,55,15)(2,12,56,29)(3,26,49,9)(4,14,50,31)(5,28,51,11)(6,16,52,25)(7,30,53,13)(8,10,54,27)(17,48,34,57)(18,62,35,45)(19,42,36,59)(20,64,37,47)(21,44,38,61)(22,58,39,41)(23,46,40,63)(24,60,33,43), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,33,21,37)(18,36,22,40)(19,39,23,35)(20,34,24,38)(25,26,29,30)(27,32,31,28)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55) );
G=PermutationGroup([(1,61,5,57),(2,62,6,58),(3,63,7,59),(4,64,8,60),(9,40,13,36),(10,33,14,37),(11,34,15,38),(12,35,16,39),(17,32,21,28),(18,25,22,29),(19,26,23,30),(20,27,24,31),(41,56,45,52),(42,49,46,53),(43,50,47,54),(44,51,48,55)], [(1,32,55,15),(2,12,56,29),(3,26,49,9),(4,14,50,31),(5,28,51,11),(6,16,52,25),(7,30,53,13),(8,10,54,27),(17,48,34,57),(18,62,35,45),(19,42,36,59),(20,64,37,47),(21,44,38,61),(22,58,39,41),(23,46,40,63),(24,60,33,43)], [(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)], [(1,4,5,8),(2,7,6,3),(9,12,13,16),(10,15,14,11),(17,33,21,37),(18,36,22,40),(19,39,23,35),(20,34,24,38),(25,26,29,30),(27,32,31,28),(41,63,45,59),(42,58,46,62),(43,61,47,57),(44,64,48,60),(49,56,53,52),(50,51,54,55)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 1 | 2 | 2 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 | 5 | 4 |
| 0 | 0 | 10 | 9 | 15 | 1 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 0 | 7 | 6 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 | 5 | 4 |
| 0 | 0 | 10 | 9 | 15 | 1 |
| 0 | 0 | 8 | 2 | 11 | 2 |
| 0 | 0 | 16 | 6 | 7 | 6 |
G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,15,2,16,0,0,0,0,2,0,16],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[7,5,0,0,0,0,7,0,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,5,15,11,7,0,0,4,1,2,6],[7,5,0,0,0,0,7,10,0,0,0,0,0,0,8,10,8,16,0,0,2,9,2,6,0,0,5,15,11,7,0,0,4,1,2,6] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | 2+ (1+4) | 2- (1+4) | D8⋊C22 |
| kernel | C42.294D4 | C2×C4⋊C8 | C4⋊SD16 | D4.D4 | C8⋊8D4 | Q8⋊Q8 | D4⋊2Q8 | C22.26C24 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{294}D_4 % in TeX
G:=Group("C4^2.294D4"); // GroupNames label
G:=SmallGroup(128,1978);
// by ID
G=gap.SmallGroup(128,1978);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations