p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.488C23, C4.742- 1+4, (C8×D4)⋊25C2, (C4×D8)⋊25C2, D4.Q8⋊8C2, C8⋊7D4⋊23C2, C4⋊C4.271D4, C8.5Q8⋊9C2, (C2×D4).243D4, C2.55(D4○D8), C8.77(C4○D4), C4⋊C8.322C22, C4⋊C4.244C23, (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, C2.D8.195C22, C4.Q8.109C22, C23.46D4⋊34C2, C4⋊D4.100C22, C23.25D4⋊11C2, 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
Generators and relations for C42.488C23
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 >
Subgroups: 360 in 183 conjugacy classes, 88 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C22×C8, C2×D8, C2×C2.D8, C23.25D4, C8×D4, C4×D8, C8⋊7D4, D4.Q8, C23.46D4, C23.19D4, C8.5Q8, C22.47C24, C42.488C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, C2×C4○D8, D4○D8, C42.488C23
►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 52 28 31)(2 49 25 32)(3 50 26 29)(4 51 27 30)(5 60 61 54)(6 57 62 55)(7 58 63 56)(8 59 64 53)(9 33 14 40)(10 34 15 37)(11 35 16 38)(12 36 13 39)(17 45 24 41)(18 46 21 42)(19 47 22 43)(20 48 23 44)
(1 56 3 54)(2 57 4 59)(5 31 7 29)(6 51 8 49)(9 43 11 41)(10 46 12 48)(13 44 15 42)(14 47 16 45)(17 40 19 38)(18 36 20 34)(21 39 23 37)(22 35 24 33)(25 55 27 53)(26 60 28 58)(30 64 32 62)(50 61 52 63)
(1 4 3 2)(5 55 7 53)(6 56 8 54)(9 15 11 13)(10 16 12 14)(17 48 19 46)(18 45 20 47)(21 41 23 43)(22 42 24 44)(25 28 27 26)(29 49 31 51)(30 50 32 52)(33 34 35 36)(37 38 39 40)(57 63 59 61)(58 64 60 62)
(1 16 28 11)(2 10 25 15)(3 14 26 9)(4 12 27 13)(5 17 61 24)(6 23 62 20)(7 19 63 22)(8 21 64 18)(29 33 50 40)(30 39 51 36)(31 35 52 38)(32 37 49 34)(41 60 45 54)(42 53 46 59)(43 58 47 56)(44 55 48 57)
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), (1,52,28,31)(2,49,25,32)(3,50,26,29)(4,51,27,30)(5,60,61,54)(6,57,62,55)(7,58,63,56)(8,59,64,53)(9,33,14,40)(10,34,15,37)(11,35,16,38)(12,36,13,39)(17,45,24,41)(18,46,21,42)(19,47,22,43)(20,48,23,44), (1,56,3,54)(2,57,4,59)(5,31,7,29)(6,51,8,49)(9,43,11,41)(10,46,12,48)(13,44,15,42)(14,47,16,45)(17,40,19,38)(18,36,20,34)(21,39,23,37)(22,35,24,33)(25,55,27,53)(26,60,28,58)(30,64,32,62)(50,61,52,63), (1,4,3,2)(5,55,7,53)(6,56,8,54)(9,15,11,13)(10,16,12,14)(17,48,19,46)(18,45,20,47)(21,41,23,43)(22,42,24,44)(25,28,27,26)(29,49,31,51)(30,50,32,52)(33,34,35,36)(37,38,39,40)(57,63,59,61)(58,64,60,62), (1,16,28,11)(2,10,25,15)(3,14,26,9)(4,12,27,13)(5,17,61,24)(6,23,62,20)(7,19,63,22)(8,21,64,18)(29,33,50,40)(30,39,51,36)(31,35,52,38)(32,37,49,34)(41,60,45,54)(42,53,46,59)(43,58,47,56)(44,55,48,57)>;
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), (1,52,28,31)(2,49,25,32)(3,50,26,29)(4,51,27,30)(5,60,61,54)(6,57,62,55)(7,58,63,56)(8,59,64,53)(9,33,14,40)(10,34,15,37)(11,35,16,38)(12,36,13,39)(17,45,24,41)(18,46,21,42)(19,47,22,43)(20,48,23,44), (1,56,3,54)(2,57,4,59)(5,31,7,29)(6,51,8,49)(9,43,11,41)(10,46,12,48)(13,44,15,42)(14,47,16,45)(17,40,19,38)(18,36,20,34)(21,39,23,37)(22,35,24,33)(25,55,27,53)(26,60,28,58)(30,64,32,62)(50,61,52,63), (1,4,3,2)(5,55,7,53)(6,56,8,54)(9,15,11,13)(10,16,12,14)(17,48,19,46)(18,45,20,47)(21,41,23,43)(22,42,24,44)(25,28,27,26)(29,49,31,51)(30,50,32,52)(33,34,35,36)(37,38,39,40)(57,63,59,61)(58,64,60,62), (1,16,28,11)(2,10,25,15)(3,14,26,9)(4,12,27,13)(5,17,61,24)(6,23,62,20)(7,19,63,22)(8,21,64,18)(29,33,50,40)(30,39,51,36)(31,35,52,38)(32,37,49,34)(41,60,45,54)(42,53,46,59)(43,58,47,56)(44,55,48,57) );
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)], [(1,52,28,31),(2,49,25,32),(3,50,26,29),(4,51,27,30),(5,60,61,54),(6,57,62,55),(7,58,63,56),(8,59,64,53),(9,33,14,40),(10,34,15,37),(11,35,16,38),(12,36,13,39),(17,45,24,41),(18,46,21,42),(19,47,22,43),(20,48,23,44)], [(1,56,3,54),(2,57,4,59),(5,31,7,29),(6,51,8,49),(9,43,11,41),(10,46,12,48),(13,44,15,42),(14,47,16,45),(17,40,19,38),(18,36,20,34),(21,39,23,37),(22,35,24,33),(25,55,27,53),(26,60,28,58),(30,64,32,62),(50,61,52,63)], [(1,4,3,2),(5,55,7,53),(6,56,8,54),(9,15,11,13),(10,16,12,14),(17,48,19,46),(18,45,20,47),(21,41,23,43),(22,42,24,44),(25,28,27,26),(29,49,31,51),(30,50,32,52),(33,34,35,36),(37,38,39,40),(57,63,59,61),(58,64,60,62)], [(1,16,28,11),(2,10,25,15),(3,14,26,9),(4,12,27,13),(5,17,61,24),(6,23,62,20),(7,19,63,22),(8,21,64,18),(29,33,50,40),(30,39,51,36),(31,35,52,38),(32,37,49,34),(41,60,45,54),(42,53,46,59),(43,58,47,56),(44,55,48,57)]])
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 |
Matrix representation of C42.488C23 ►in 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] >;
C42.488C23 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