p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.465C23, C4.452+ 1+4, (C8×D4)⋊18C2, Q8.Q8⋊5C2, C8⋊8D4⋊38C2, C4⋊C4.264D4, (C4×Q16)⋊12C2, (C2×D4).235D4, (C4×C8).84C22, Q8⋊5D4.2C2, Q8.D4⋊8C2, C2.45(Q8○D8), D4.7D4⋊10C2, C8.18D4⋊13C2, C4⋊C4.229C23, C4⋊C8.318C22, (C2×C4).492C24, (C2×C8).184C23, Q8.17(C4○D4), C22⋊Q16⋊10C2, C22⋊C4.104D4, C23.108(C2×D4), C22.D8⋊8C2, C2.D8.54C22, (C2×D4).223C23, (C4×D4).333C22, C22.12(C4○D8), C4⋊D4.73C22, C23.20D4⋊4C2, (C2×Q8).209C23, (C4×Q8).149C22, C2.128(D4⋊5D4), C4.Q8.100C22, C22⋊Q8.72C22, D4⋊C4.12C22, C23.24D4⋊11C2, C22⋊C8.201C22, (C22×C8).161C22, (C2×Q16).133C22, Q8⋊C4.12C22, (C2×SD16).96C22, C4.4D4.60C22, C22.752(C22×D4), C42.C2.35C22, C42.78C22⋊5C2, (C22×C4).1136C23, C22.46C24⋊2C2, (C22×Q8).339C22, C42⋊C2.180C22, C2.60(C2×C4○D8), C4.217(C2×C4○D4), (C2×C4).169(C2×D4), (C2×Q8⋊C4)⋊42C2, C22⋊C4○(Q8⋊C4), (C2×C4⋊C4).662C22, (C2×C4○D4).198C22, SmallGroup(128,2032)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.465C23
G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=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 193 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C22×C8, C2×SD16, C2×Q16, C22×Q8, C2×C4○D4, C2×Q8⋊C4, C23.24D4, C8×D4, C4×Q16, C22⋊Q16, D4.7D4, Q8.D4, C8⋊8D4, C8.18D4, Q8.Q8, C22.D8, C23.20D4, C42.78C22, Q8⋊5D4, C22.46C24, C42.465C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C4○D8, Q8○D8, C42.465C23
►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 41 12 50)(2 42 9 51)(3 43 10 52)(4 44 11 49)(5 34 26 55)(6 35 27 56)(7 36 28 53)(8 33 25 54)(13 21 17 46)(14 22 18 47)(15 23 19 48)(16 24 20 45)(29 37 61 60)(30 38 62 57)(31 39 63 58)(32 40 64 59)
(1 55 10 36)(2 33 11 56)(3 53 12 34)(4 35 9 54)(5 43 28 50)(6 51 25 44)(7 41 26 52)(8 49 27 42)(13 60 19 39)(14 40 20 57)(15 58 17 37)(16 38 18 59)(21 61 48 31)(22 32 45 62)(23 63 46 29)(24 30 47 64)
(1 16 12 20)(2 13 9 17)(3 14 10 18)(4 15 11 19)(5 57 26 38)(6 58 27 39)(7 59 28 40)(8 60 25 37)(21 42 46 51)(22 43 47 52)(23 44 48 49)(24 41 45 50)(29 33 61 54)(30 34 62 55)(31 35 63 56)(32 36 64 53)
(1 20 12 16)(2 15 9 19)(3 18 10 14)(4 13 11 17)(5 64 26 32)(6 31 27 63)(7 62 28 30)(8 29 25 61)(21 49 46 44)(22 43 47 52)(23 51 48 42)(24 41 45 50)(33 37 54 60)(34 59 55 40)(35 39 56 58)(36 57 53 38)
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), (1,41,12,50)(2,42,9,51)(3,43,10,52)(4,44,11,49)(5,34,26,55)(6,35,27,56)(7,36,28,53)(8,33,25,54)(13,21,17,46)(14,22,18,47)(15,23,19,48)(16,24,20,45)(29,37,61,60)(30,38,62,57)(31,39,63,58)(32,40,64,59), (1,55,10,36)(2,33,11,56)(3,53,12,34)(4,35,9,54)(5,43,28,50)(6,51,25,44)(7,41,26,52)(8,49,27,42)(13,60,19,39)(14,40,20,57)(15,58,17,37)(16,38,18,59)(21,61,48,31)(22,32,45,62)(23,63,46,29)(24,30,47,64), (1,16,12,20)(2,13,9,17)(3,14,10,18)(4,15,11,19)(5,57,26,38)(6,58,27,39)(7,59,28,40)(8,60,25,37)(21,42,46,51)(22,43,47,52)(23,44,48,49)(24,41,45,50)(29,33,61,54)(30,34,62,55)(31,35,63,56)(32,36,64,53), (1,20,12,16)(2,15,9,19)(3,18,10,14)(4,13,11,17)(5,64,26,32)(6,31,27,63)(7,62,28,30)(8,29,25,61)(21,49,46,44)(22,43,47,52)(23,51,48,42)(24,41,45,50)(33,37,54,60)(34,59,55,40)(35,39,56,58)(36,57,53,38)>;
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), (1,41,12,50)(2,42,9,51)(3,43,10,52)(4,44,11,49)(5,34,26,55)(6,35,27,56)(7,36,28,53)(8,33,25,54)(13,21,17,46)(14,22,18,47)(15,23,19,48)(16,24,20,45)(29,37,61,60)(30,38,62,57)(31,39,63,58)(32,40,64,59), (1,55,10,36)(2,33,11,56)(3,53,12,34)(4,35,9,54)(5,43,28,50)(6,51,25,44)(7,41,26,52)(8,49,27,42)(13,60,19,39)(14,40,20,57)(15,58,17,37)(16,38,18,59)(21,61,48,31)(22,32,45,62)(23,63,46,29)(24,30,47,64), (1,16,12,20)(2,13,9,17)(3,14,10,18)(4,15,11,19)(5,57,26,38)(6,58,27,39)(7,59,28,40)(8,60,25,37)(21,42,46,51)(22,43,47,52)(23,44,48,49)(24,41,45,50)(29,33,61,54)(30,34,62,55)(31,35,63,56)(32,36,64,53), (1,20,12,16)(2,15,9,19)(3,18,10,14)(4,13,11,17)(5,64,26,32)(6,31,27,63)(7,62,28,30)(8,29,25,61)(21,49,46,44)(22,43,47,52)(23,51,48,42)(24,41,45,50)(33,37,54,60)(34,59,55,40)(35,39,56,58)(36,57,53,38) );
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)], [(1,41,12,50),(2,42,9,51),(3,43,10,52),(4,44,11,49),(5,34,26,55),(6,35,27,56),(7,36,28,53),(8,33,25,54),(13,21,17,46),(14,22,18,47),(15,23,19,48),(16,24,20,45),(29,37,61,60),(30,38,62,57),(31,39,63,58),(32,40,64,59)], [(1,55,10,36),(2,33,11,56),(3,53,12,34),(4,35,9,54),(5,43,28,50),(6,51,25,44),(7,41,26,52),(8,49,27,42),(13,60,19,39),(14,40,20,57),(15,58,17,37),(16,38,18,59),(21,61,48,31),(22,32,45,62),(23,63,46,29),(24,30,47,64)], [(1,16,12,20),(2,13,9,17),(3,14,10,18),(4,15,11,19),(5,57,26,38),(6,58,27,39),(7,59,28,40),(8,60,25,37),(21,42,46,51),(22,43,47,52),(23,44,48,49),(24,41,45,50),(29,33,61,54),(30,34,62,55),(31,35,63,56),(32,36,64,53)], [(1,20,12,16),(2,15,9,19),(3,18,10,14),(4,13,11,17),(5,64,26,32),(6,31,27,63),(7,62,28,30),(8,29,25,61),(21,49,46,44),(22,43,47,52),(23,51,48,42),(24,41,45,50),(33,37,54,60),(34,59,55,40),(35,39,56,58),(36,57,53,38)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 4M | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2+ 1+4 | Q8○D8 |
| kernel | C42.465C23 | C2×Q8⋊C4 | C23.24D4 | C8×D4 | C4×Q16 | C22⋊Q16 | D4.7D4 | Q8.D4 | C8⋊8D4 | C8.18D4 | Q8.Q8 | C22.D8 | C23.20D4 | C42.78C22 | Q8⋊5D4 | C22.46C24 | C22⋊C4 | C4⋊C4 | C2×D4 | Q8 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.465C23 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 13 |
| 0 | 0 | 9 | 16 |
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 11 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 16 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,1,9,0,0,13,16],[16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,11,6,0,0,0,0,4,0,0,0,1,13],[13,4,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,9,0,0,0,16] >;
C42.465C23 in GAP, Magma, Sage, TeX
C_4^2._{465}C_2^3 % in TeX
G:=Group("C4^2.465C2^3"); // GroupNames label
G:=SmallGroup(128,2032);
// by ID
G=gap.SmallGroup(128,2032);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,352,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2*b^2,d^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