p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.694C23, C4.1732+ (1+4), (C8×D4)⋊48C2, C4⋊Q8.32C4, C8⋊6D4⋊43C2, C4.40(C8○D4), C4⋊D4.27C4, C4⋊1D4.19C4, C4⋊C8.366C22, (C4×C8).338C22, C42.224(C2×C4), (C2×C8).437C23, (C2×C4).676C24, C4.4D4.20C4, (C4×D4).302C22, C23.43(C22×C4), (C22×C8).94C22, C42.12C4⋊54C2, C22⋊C8.235C22, (C2×C42).783C22, C22.200(C23×C4), (C22×C4).943C23, C2.50(C22.11C24), (C2×M4(2)).246C22, C22.26C24.27C2, C2.30(C2×C8○D4), C4⋊C4.169(C2×C4), (C2×D4).184(C2×C4), C22⋊C4.44(C2×C4), (C2×C4).82(C22×C4), (C2×Q8).125(C2×C4), (C22×C8)⋊C2⋊35C2, (C22×C4).356(C2×C4), (C2×C4○D4).96C22, SmallGroup(128,1711)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 332 in 205 conjugacy classes, 128 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×15], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×14], Q8 [×2], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.12C4 [×2], C8×D4 [×4], C8⋊6D4 [×4], C22.26C24, C42.694C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×4], C23×C4, 2+ (1+4) [×2], C22.11C24, C2×C8○D4 [×2], C42.694C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=1, c2=b, e2=a2, ab=ba, ac=ca, dad=a-1b2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >
►On 64 points
(1 41 55 57)(2 42 56 58)(3 43 49 59)(4 44 50 60)(5 45 51 61)(6 46 52 62)(7 47 53 63)(8 48 54 64)(9 28 38 20)(10 29 39 21)(11 30 40 22)(12 31 33 23)(13 32 34 24)(14 25 35 17)(15 26 36 18)(16 27 37 19)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(1 33 55 12)(2 13 56 34)(3 35 49 14)(4 15 50 36)(5 37 51 16)(6 9 52 38)(7 39 53 10)(8 11 54 40)(17 59 25 43)(18 44 26 60)(19 61 27 45)(20 46 28 62)(21 63 29 47)(22 48 30 64)(23 57 31 41)(24 42 32 58)
G:=sub<Sym(64)| (1,41,55,57)(2,42,56,58)(3,43,49,59)(4,44,50,60)(5,45,51,61)(6,46,52,62)(7,47,53,63)(8,48,54,64)(9,28,38,20)(10,29,39,21)(11,30,40,22)(12,31,33,23)(13,32,34,24)(14,25,35,17)(15,26,36,18)(16,27,37,19), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,33,55,12)(2,13,56,34)(3,35,49,14)(4,15,50,36)(5,37,51,16)(6,9,52,38)(7,39,53,10)(8,11,54,40)(17,59,25,43)(18,44,26,60)(19,61,27,45)(20,46,28,62)(21,63,29,47)(22,48,30,64)(23,57,31,41)(24,42,32,58)>;
G:=Group( (1,41,55,57)(2,42,56,58)(3,43,49,59)(4,44,50,60)(5,45,51,61)(6,46,52,62)(7,47,53,63)(8,48,54,64)(9,28,38,20)(10,29,39,21)(11,30,40,22)(12,31,33,23)(13,32,34,24)(14,25,35,17)(15,26,36,18)(16,27,37,19), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,33,55,12)(2,13,56,34)(3,35,49,14)(4,15,50,36)(5,37,51,16)(6,9,52,38)(7,39,53,10)(8,11,54,40)(17,59,25,43)(18,44,26,60)(19,61,27,45)(20,46,28,62)(21,63,29,47)(22,48,30,64)(23,57,31,41)(24,42,32,58) );
G=PermutationGroup([(1,41,55,57),(2,42,56,58),(3,43,49,59),(4,44,50,60),(5,45,51,61),(6,46,52,62),(7,47,53,63),(8,48,54,64),(9,28,38,20),(10,29,39,21),(11,30,40,22),(12,31,33,23),(13,32,34,24),(14,25,35,17),(15,26,36,18),(16,27,37,19)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(1,33,55,12),(2,13,56,34),(3,35,49,14),(4,15,50,36),(5,37,51,16),(6,9,52,38),(7,39,53,10),(8,11,54,40),(17,59,25,43),(18,44,26,60),(19,61,27,45),(20,46,28,62),(21,63,29,47),(22,48,30,64),(23,57,31,41),(24,42,32,58)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[15,0,0,0,0,15,0,0,0,0,2,0,0,0,0,15],[0,13,0,0,4,0,0,0,0,0,16,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4Q | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8○D4 | 2+ (1+4) |
| kernel | C42.694C23 | (C22×C8)⋊C2 | C42.12C4 | C8×D4 | C8⋊6D4 | C22.26C24 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | C4 | C4 |
| # reps | 1 | 4 | 2 | 4 | 4 | 1 | 8 | 4 | 2 | 2 | 16 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{694}C_2^3 % in TeX
G:=Group("C4^2.694C2^3"); // GroupNames label
G:=SmallGroup(128,1711);
// by ID
G=gap.SmallGroup(128,1711);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,521,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=b,e^2=a^2,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=a^2*d>;
// generators/relations