p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.294C23, (C8×D4)⋊43C2, (C8×Q8)⋊31C2, C8⋊9D4⋊37C2, C8⋊6D4⋊37C2, C8⋊4Q8⋊37C2, C4.39(C8○D4), C8.87(C4○D4), C4⋊D4.23C4, C22⋊Q8.23C4, C4⋊C8.364C22, (C4×M4(2))⋊37C2, (C2×C4).666C24, C42⋊2C2.3C4, (C4×C8).335C22, C42.217(C2×C4), (C2×C8).614C23, C4.4D4.19C4, C42.C2.19C4, C8○2M4(2)⋊36C2, (C4×D4).295C22, C23.40(C22×C4), (C4×Q8).280C22, C22.D4.7C4, C8⋊C4.175C22, C42.12C4⋊51C2, C22⋊C8.142C22, C2.24(Q8○M4(2)), C22.191(C23×C4), (C22×C8).448C22, (C2×C42).776C22, (C22×C4).936C23, C42.7C22⋊25C2, C42⋊C2.309C22, (C2×M4(2)).368C22, C23.36C23.15C2, C8⋊C4○(C4⋊C8), C2.48(C4×C4○D4), C2.26(C2×C8○D4), C4⋊C4.166(C2×C4), C4.317(C2×C4○D4), (C2×D4).142(C2×C4), C22⋊C4.18(C2×C4), (C2×Q8).165(C2×C4), (C22×C4).350(C2×C4), (C2×C4).273(C22×C4), SmallGroup(128,1701)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 252 in 185 conjugacy classes, 130 normal (52 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×9], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×4], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C4×C8 [×4], C8⋊C4 [×2], C8⋊C4 [×2], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×4], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C4×M4(2), C8○2M4(2) [×2], C42.12C4, C42.7C22 [×2], C8×D4, C8⋊9D4 [×2], C8⋊6D4, C8⋊6D4 [×2], C8×Q8, C8⋊4Q8, C23.36C23, C42.294C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C4×C4○D4, C2×C8○D4, Q8○M4(2), C42.294C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b, e2=a2, ab=ba, cac-1=a-1b2, dad-1=eae-1=ab2, bc=cb, bd=db, be=eb, dcd-1=a2c, ce=ec, de=ed >
►On 64 points
(1 37 57 48)(2 34 58 45)(3 39 59 42)(4 36 60 47)(5 33 61 44)(6 38 62 41)(7 35 63 46)(8 40 64 43)(9 55 32 22)(10 52 25 19)(11 49 26 24)(12 54 27 21)(13 51 28 18)(14 56 29 23)(15 53 30 20)(16 50 31 17)
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)
(1 48 5 44)(2 38 6 34)(3 42 7 46)(4 40 8 36)(9 18 13 22)(10 52 14 56)(11 20 15 24)(12 54 16 50)(17 27 21 31)(19 29 23 25)(26 53 30 49)(28 55 32 51)(33 57 37 61)(35 59 39 63)(41 62 45 58)(43 64 47 60)
(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 21 57 54)(2 22 58 55)(3 23 59 56)(4 24 60 49)(5 17 61 50)(6 18 62 51)(7 19 63 52)(8 20 64 53)(9 41 32 38)(10 42 25 39)(11 43 26 40)(12 44 27 33)(13 45 28 34)(14 46 29 35)(15 47 30 36)(16 48 31 37)
G:=sub<Sym(64)| (1,37,57,48)(2,34,58,45)(3,39,59,42)(4,36,60,47)(5,33,61,44)(6,38,62,41)(7,35,63,46)(8,40,64,43)(9,55,32,22)(10,52,25,19)(11,49,26,24)(12,54,27,21)(13,51,28,18)(14,56,29,23)(15,53,30,20)(16,50,31,17), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45), (1,48,5,44)(2,38,6,34)(3,42,7,46)(4,40,8,36)(9,18,13,22)(10,52,14,56)(11,20,15,24)(12,54,16,50)(17,27,21,31)(19,29,23,25)(26,53,30,49)(28,55,32,51)(33,57,37,61)(35,59,39,63)(41,62,45,58)(43,64,47,60), (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), (1,21,57,54)(2,22,58,55)(3,23,59,56)(4,24,60,49)(5,17,61,50)(6,18,62,51)(7,19,63,52)(8,20,64,53)(9,41,32,38)(10,42,25,39)(11,43,26,40)(12,44,27,33)(13,45,28,34)(14,46,29,35)(15,47,30,36)(16,48,31,37)>;
G:=Group( (1,37,57,48)(2,34,58,45)(3,39,59,42)(4,36,60,47)(5,33,61,44)(6,38,62,41)(7,35,63,46)(8,40,64,43)(9,55,32,22)(10,52,25,19)(11,49,26,24)(12,54,27,21)(13,51,28,18)(14,56,29,23)(15,53,30,20)(16,50,31,17), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45), (1,48,5,44)(2,38,6,34)(3,42,7,46)(4,40,8,36)(9,18,13,22)(10,52,14,56)(11,20,15,24)(12,54,16,50)(17,27,21,31)(19,29,23,25)(26,53,30,49)(28,55,32,51)(33,57,37,61)(35,59,39,63)(41,62,45,58)(43,64,47,60), (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), (1,21,57,54)(2,22,58,55)(3,23,59,56)(4,24,60,49)(5,17,61,50)(6,18,62,51)(7,19,63,52)(8,20,64,53)(9,41,32,38)(10,42,25,39)(11,43,26,40)(12,44,27,33)(13,45,28,34)(14,46,29,35)(15,47,30,36)(16,48,31,37) );
G=PermutationGroup([(1,37,57,48),(2,34,58,45),(3,39,59,42),(4,36,60,47),(5,33,61,44),(6,38,62,41),(7,35,63,46),(8,40,64,43),(9,55,32,22),(10,52,25,19),(11,49,26,24),(12,54,27,21),(13,51,28,18),(14,56,29,23),(15,53,30,20),(16,50,31,17)], [(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,26,13,30),(10,27,14,31),(11,28,15,32),(12,29,16,25),(17,52,21,56),(18,53,22,49),(19,54,23,50),(20,55,24,51),(33,46,37,42),(34,47,38,43),(35,48,39,44),(36,41,40,45)], [(1,48,5,44),(2,38,6,34),(3,42,7,46),(4,40,8,36),(9,18,13,22),(10,52,14,56),(11,20,15,24),(12,54,16,50),(17,27,21,31),(19,29,23,25),(26,53,30,49),(28,55,32,51),(33,57,37,61),(35,59,39,63),(41,62,45,58),(43,64,47,60)], [(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)], [(1,21,57,54),(2,22,58,55),(3,23,59,56),(4,24,60,49),(5,17,61,50),(6,18,62,51),(7,19,63,52),(8,20,64,53),(9,41,32,38),(10,42,25,39),(11,43,26,40),(12,44,27,33),(13,45,28,34),(14,46,29,35),(15,47,30,36),(16,48,31,37)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 0 | 4 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 15 | 4 |
| 0 | 0 | 0 | 2 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,4,4,0,0,9,13],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,16,0,0,0,0,0,13,0,0,0,8,4],[0,1,0,0,16,0,0,0,0,0,15,0,0,0,4,2],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,15,16] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4S | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | C42.294C23 | C4×M4(2) | C8○2M4(2) | C42.12C4 | C42.7C22 | C8×D4 | C8⋊9D4 | C8⋊6D4 | C8×Q8 | C8⋊4Q8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{294}C_2^3 % in TeX
G:=Group("C4^2.294C2^3"); // GroupNames label
G:=SmallGroup(128,1701);
// by ID
G=gap.SmallGroup(128,1701);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,521,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2*b,e^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=e*a*e^-1=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,c*e=e*c,d*e=e*d>;
// generators/relations