p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.506C23, C4.272- 1+4, (C8×Q8)⋊13C2, C4⋊C4.279D4, (C4×Q16)⋊17C2, Q8⋊3Q8⋊4C2, D4.Q8⋊10C2, Q8⋊Q8⋊46C2, (C4×SD16)⋊48C2, C4.92(C4○D8), Q8.D4⋊9C2, (C2×Q8).185D4, C4.4D8.8C2, C4⋊C8.326C22, C4⋊C4.433C23, (C2×C4).557C24, (C2×C8).370C23, (C4×C8).281C22, Q8.35(C4○D4), C4⋊SD16.12C2, C4⋊Q8.186C22, C2.65(Q8⋊5D4), C2.99(D4○SD16), (C4×D4).196C22, (C2×D4).269C23, C4⋊1D4.97C22, (C4×Q8).188C22, (C2×Q8).403C23, C2.D8.203C22, C4.Q8.177C22, (C2×Q16).143C22, C4.4D4.77C22, C22.817(C22×D4), C42.C2.62C22, D4⋊C4.128C22, Q8⋊C4.143C22, (C2×SD16).172C22, C42.78C22⋊20C2, C22.53C24.3C2, C2.75(C2×C4○D8), C4.258(C2×C4○D4), (C2×C4).177(C2×D4), SmallGroup(128,2097)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.506C23
G = < a,b,c,d,e | a4=b4=1, c2=e2=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, de=ed >
Subgroups: 320 in 174 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×Q16, C4×SD16, C4×Q16, C8×Q8, C4⋊SD16, Q8.D4, Q8⋊Q8, D4.Q8, C4.4D8, C42.78C22, Q8⋊3Q8, C22.53C24, C42.506C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C2×C4○D8, D4○SD16, C42.506C23
►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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(1 55 25 59)(2 56 26 60)(3 53 27 57)(4 54 28 58)(5 38 9 36)(6 39 10 33)(7 40 11 34)(8 37 12 35)(13 32 62 44)(14 29 63 41)(15 30 64 42)(16 31 61 43)(17 49 23 47)(18 50 24 48)(19 51 21 45)(20 52 22 46)
(1 57 27 55)(2 60 28 54)(3 59 25 53)(4 58 26 56)(5 44 11 30)(6 43 12 29)(7 42 9 32)(8 41 10 31)(13 36 64 40)(14 35 61 39)(15 34 62 38)(16 33 63 37)(17 51 21 47)(18 50 22 46)(19 49 23 45)(20 52 24 48)
(1 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 60 9 56)(6 57 10 53)(7 58 11 54)(8 59 12 55)(13 52 62 46)(14 49 63 47)(15 50 64 48)(16 51 61 45)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)
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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,55,25,59)(2,56,26,60)(3,53,27,57)(4,54,28,58)(5,38,9,36)(6,39,10,33)(7,40,11,34)(8,37,12,35)(13,32,62,44)(14,29,63,41)(15,30,64,42)(16,31,61,43)(17,49,23,47)(18,50,24,48)(19,51,21,45)(20,52,22,46), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,60,9,56)(6,57,10,53)(7,58,11,54)(8,59,12,55)(13,52,62,46)(14,49,63,47)(15,50,64,48)(16,51,61,45)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40)>;
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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,55,25,59)(2,56,26,60)(3,53,27,57)(4,54,28,58)(5,38,9,36)(6,39,10,33)(7,40,11,34)(8,37,12,35)(13,32,62,44)(14,29,63,41)(15,30,64,42)(16,31,61,43)(17,49,23,47)(18,50,24,48)(19,51,21,45)(20,52,22,46), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,60,9,56)(6,57,10,53)(7,58,11,54)(8,59,12,55)(13,52,62,46)(14,49,63,47)(15,50,64,48)(16,51,61,45)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40) );
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)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,37,41,35),(30,38,42,36),(31,39,43,33),(32,40,44,34),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(1,55,25,59),(2,56,26,60),(3,53,27,57),(4,54,28,58),(5,38,9,36),(6,39,10,33),(7,40,11,34),(8,37,12,35),(13,32,62,44),(14,29,63,41),(15,30,64,42),(16,31,61,43),(17,49,23,47),(18,50,24,48),(19,51,21,45),(20,52,22,46)], [(1,57,27,55),(2,60,28,54),(3,59,25,53),(4,58,26,56),(5,44,11,30),(6,43,12,29),(7,42,9,32),(8,41,10,31),(13,36,64,40),(14,35,61,39),(15,34,62,38),(16,33,63,37),(17,51,21,47),(18,50,22,46),(19,49,23,45),(20,52,24,48)], [(1,41,25,29),(2,42,26,30),(3,43,27,31),(4,44,28,32),(5,60,9,56),(6,57,10,53),(7,58,11,54),(8,59,12,55),(13,52,62,46),(14,49,63,47),(15,50,64,48),(16,51,61,45),(17,35,23,37),(18,36,24,38),(19,33,21,39),(20,34,22,40)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4O | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 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 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | 2- 1+4 | D4○SD16 |
| kernel | C42.506C23 | C4×SD16 | C4×Q16 | C8×Q8 | C4⋊SD16 | Q8.D4 | Q8⋊Q8 | D4.Q8 | C4.4D8 | C42.78C22 | Q8⋊3Q8 | C22.53C24 | C4⋊C4 | C2×Q8 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.506C23 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 13 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 5 | 7 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[0,13,0,0,4,0,0,0,0,0,4,4,0,0,0,13],[0,4,0,0,4,0,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,10,5,0,0,7,7] >;
C42.506C23 in GAP, Magma, Sage, TeX
C_4^2._{506}C_2^3 % in TeX
G:=Group("C4^2.506C2^3"); // GroupNames label
G:=SmallGroup(128,2097);
// by ID
G=gap.SmallGroup(128,2097);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=e^2=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations