p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.488C23, C4.742- (1+4), (C4×D8)⋊25C2, (C8×D4)⋊25C2, D4.Q8⋊8C2, C8⋊7D4⋊23C2, C4⋊C4.271D4, C8.5Q8⋊9C2, (C2×D4).243D4, C8.77(C4○D4), C2.55(D4○D8), C4⋊C4.244C23, C4⋊C8.322C22, (C4×C8).122C22, (C2×C8).198C23, (C2×C4).531C24, C22⋊C4.115D4, C23.116(C2×D4), C2.84(D4⋊6D4), (C4×D4).344C22, (C2×D8).143C22, (C2×D4).251C23, C22.15(C4○D8), C23.19D4⋊9C2, C4.Q8.109C22, C2.D8.195C22, C4⋊D4.100C22, C23.25D4⋊11C2, C23.46D4⋊34C2, C22⋊C8.209C22, (C22×C8).198C22, C22.791(C22×D4), C42.C2.47C22, D4⋊C4.125C22, C22.47C24⋊8C2, (C22×C4).1163C23, C42⋊C2.202C22, (C2×C2.D8)⋊31C2, C2.68(C2×C4○D8), C4.113(C2×C4○D4), (C2×C4).933(C2×D4), (C2×C4⋊C4).683C22, SmallGroup(128,2071)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 360 in 183 conjugacy classes, 88 normal (44 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×12], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], D8 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×4], C4×C8, C22⋊C8 [×2], D4⋊C4 [×6], C4⋊C8, C4.Q8 [×4], C2.D8 [×3], C2.D8 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22.D4 [×2], C42.C2 [×2], C42⋊2C2 [×2], C22×C8 [×2], C2×D8, C2×C2.D8, C23.25D4, C8×D4, C4×D8, C8⋊7D4 [×2], D4.Q8 [×2], C23.46D4 [×2], C23.19D4 [×2], C8.5Q8, C22.47C24 [×2], C42.488C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, C2×C4○D8, D4○D8, C42.488C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=d2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
►On 64 points
(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 36 12 31)(2 33 9 32)(3 34 10 29)(4 35 11 30)(5 58 18 42)(6 59 19 43)(7 60 20 44)(8 57 17 41)(13 52 28 39)(14 49 25 40)(15 50 26 37)(16 51 27 38)(21 54 61 48)(22 55 62 45)(23 56 63 46)(24 53 64 47)
(1 46 3 48)(2 55 4 53)(5 39 7 37)(6 51 8 49)(9 45 11 47)(10 54 12 56)(13 44 15 42)(14 59 16 57)(17 40 19 38)(18 52 20 50)(21 31 23 29)(22 35 24 33)(25 43 27 41)(26 58 28 60)(30 64 32 62)(34 61 36 63)
(1 4 3 2)(5 57 7 59)(6 58 8 60)(9 12 11 10)(13 25 15 27)(14 26 16 28)(17 44 19 42)(18 41 20 43)(21 45 23 47)(22 46 24 48)(29 33 31 35)(30 34 32 36)(37 38 39 40)(49 50 51 52)(53 61 55 63)(54 62 56 64)
(1 27 12 16)(2 15 9 26)(3 25 10 14)(4 13 11 28)(5 64 18 24)(6 23 19 63)(7 62 20 22)(8 21 17 61)(29 49 34 40)(30 39 35 52)(31 51 36 38)(32 37 33 50)(41 48 57 54)(42 53 58 47)(43 46 59 56)(44 55 60 45)
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), (1,36,12,31)(2,33,9,32)(3,34,10,29)(4,35,11,30)(5,58,18,42)(6,59,19,43)(7,60,20,44)(8,57,17,41)(13,52,28,39)(14,49,25,40)(15,50,26,37)(16,51,27,38)(21,54,61,48)(22,55,62,45)(23,56,63,46)(24,53,64,47), (1,46,3,48)(2,55,4,53)(5,39,7,37)(6,51,8,49)(9,45,11,47)(10,54,12,56)(13,44,15,42)(14,59,16,57)(17,40,19,38)(18,52,20,50)(21,31,23,29)(22,35,24,33)(25,43,27,41)(26,58,28,60)(30,64,32,62)(34,61,36,63), (1,4,3,2)(5,57,7,59)(6,58,8,60)(9,12,11,10)(13,25,15,27)(14,26,16,28)(17,44,19,42)(18,41,20,43)(21,45,23,47)(22,46,24,48)(29,33,31,35)(30,34,32,36)(37,38,39,40)(49,50,51,52)(53,61,55,63)(54,62,56,64), (1,27,12,16)(2,15,9,26)(3,25,10,14)(4,13,11,28)(5,64,18,24)(6,23,19,63)(7,62,20,22)(8,21,17,61)(29,49,34,40)(30,39,35,52)(31,51,36,38)(32,37,33,50)(41,48,57,54)(42,53,58,47)(43,46,59,56)(44,55,60,45)>;
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), (1,36,12,31)(2,33,9,32)(3,34,10,29)(4,35,11,30)(5,58,18,42)(6,59,19,43)(7,60,20,44)(8,57,17,41)(13,52,28,39)(14,49,25,40)(15,50,26,37)(16,51,27,38)(21,54,61,48)(22,55,62,45)(23,56,63,46)(24,53,64,47), (1,46,3,48)(2,55,4,53)(5,39,7,37)(6,51,8,49)(9,45,11,47)(10,54,12,56)(13,44,15,42)(14,59,16,57)(17,40,19,38)(18,52,20,50)(21,31,23,29)(22,35,24,33)(25,43,27,41)(26,58,28,60)(30,64,32,62)(34,61,36,63), (1,4,3,2)(5,57,7,59)(6,58,8,60)(9,12,11,10)(13,25,15,27)(14,26,16,28)(17,44,19,42)(18,41,20,43)(21,45,23,47)(22,46,24,48)(29,33,31,35)(30,34,32,36)(37,38,39,40)(49,50,51,52)(53,61,55,63)(54,62,56,64), (1,27,12,16)(2,15,9,26)(3,25,10,14)(4,13,11,28)(5,64,18,24)(6,23,19,63)(7,62,20,22)(8,21,17,61)(29,49,34,40)(30,39,35,52)(31,51,36,38)(32,37,33,50)(41,48,57,54)(42,53,58,47)(43,46,59,56)(44,55,60,45) );
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)], [(1,36,12,31),(2,33,9,32),(3,34,10,29),(4,35,11,30),(5,58,18,42),(6,59,19,43),(7,60,20,44),(8,57,17,41),(13,52,28,39),(14,49,25,40),(15,50,26,37),(16,51,27,38),(21,54,61,48),(22,55,62,45),(23,56,63,46),(24,53,64,47)], [(1,46,3,48),(2,55,4,53),(5,39,7,37),(6,51,8,49),(9,45,11,47),(10,54,12,56),(13,44,15,42),(14,59,16,57),(17,40,19,38),(18,52,20,50),(21,31,23,29),(22,35,24,33),(25,43,27,41),(26,58,28,60),(30,64,32,62),(34,61,36,63)], [(1,4,3,2),(5,57,7,59),(6,58,8,60),(9,12,11,10),(13,25,15,27),(14,26,16,28),(17,44,19,42),(18,41,20,43),(21,45,23,47),(22,46,24,48),(29,33,31,35),(30,34,32,36),(37,38,39,40),(49,50,51,52),(53,61,55,63),(54,62,56,64)], [(1,27,12,16),(2,15,9,26),(3,25,10,14),(4,13,11,28),(5,64,18,24),(6,23,19,63),(7,62,20,22),(8,21,17,61),(29,49,34,40),(30,39,35,52),(31,51,36,38),(32,37,33,50),(41,48,57,54),(42,53,58,47),(43,46,59,56),(44,55,60,45)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 13 | 4 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,13,13,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,5,0,0,5,5,0,0,0,0,13,0,0,0,8,4],[4,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,4,0,0,9,13] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2- (1+4) | D4○D8 |
| kernel | C42.488C23 | C2×C2.D8 | C23.25D4 | C8×D4 | C4×D8 | C8⋊7D4 | D4.Q8 | C23.46D4 | C23.19D4 | C8.5Q8 | C22.47C24 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{488}C_2^3 % in TeX
G:=Group("C4^2.488C2^3"); // GroupNames label
G:=SmallGroup(128,2071);
// by ID
G=gap.SmallGroup(128,2071);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,436,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations