p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.681C23, C8○(C4⋊Q8), (C2×C8)⋊34D4, (C8×D4)⋊37C2, C4⋊1(C8○D4), C8○(C4⋊1D4), C8○(C8⋊6D4), C4.40(C4×D4), C4⋊Q8.38C4, C8○2(C4⋊D4), C8⋊6D4⋊50C2, C8○(C4.4D4), C8.144(C2×D4), C22.2(C4×D4), C4⋊1D4.22C4, C4⋊D4.35C4, C4⋊C8.355C22, C8○(C4⋊M4(2)), (C2×C4).650C24, C42.283(C2×C4), (C4×C8).379C22, (C2×C8).403C23, C4.4D4.27C4, C4.196(C22×D4), C4⋊M4(2)⋊42C2, (C4×D4).287C22, C23.34(C22×C4), C22⋊C8.230C22, C8○(C22.26C24), C22.177(C23×C4), (C22×C8).433C22, (C22×C4).917C23, (C2×C42).1110C22, (C2×M4(2)).347C22, C22.26C24.52C2, (C2×C4×C8)⋊43C2, C2.48(C2×C4×D4), (C2×C8○D4)⋊21C2, (C2×C8)○(C4⋊1D4), (C2×C8)○(C8⋊6D4), (C2×C8)○(C4⋊D4), C2.17(C2×C8○D4), C4⋊C4.160(C2×C4), C4.301(C2×C4○D4), (C2×C4).843(C2×D4), (C2×D4).172(C2×C4), C22⋊C4.35(C2×C4), (C2×Q8).154(C2×C4), C8○2((C22×C8)⋊C2), (C22×C8)⋊C2⋊39C2, (C2×C4).683(C4○D4), (C2×C8)○(C4⋊M4(2)), (C22×C4).417(C2×C4), (C2×C4).293(C22×C4), (C2×C4○D4).285C22, (C2×C8)○(C22.26C24), SmallGroup(128,1663)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.681C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b-1, ab=ba, cac-1=a-1b2, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ece=b2c, de=ed >
Subgroups: 364 in 250 conjugacy classes, 144 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C2×C4×C8, (C22×C8)⋊C2, C4⋊M4(2), C8×D4, C8⋊6D4, C22.26C24, C2×C8○D4, C42.681C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C8○D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4, C42.681C23
►On 64 points
(1 41 28 24)(2 42 29 17)(3 43 30 18)(4 44 31 19)(5 45 32 20)(6 46 25 21)(7 47 26 22)(8 48 27 23)(9 33 55 58)(10 34 56 59)(11 35 49 60)(12 36 50 61)(13 37 51 62)(14 38 52 63)(15 39 53 64)(16 40 54 57)
(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 53 13 49)(10 54 14 50)(11 55 15 51)(12 56 16 52)(17 48 21 44)(18 41 22 45)(19 42 23 46)(20 43 24 47)(33 64 37 60)(34 57 38 61)(35 58 39 62)(36 59 40 63)
(1 15 5 11)(2 50 6 54)(3 9 7 13)(4 52 8 56)(10 31 14 27)(12 25 16 29)(17 57 21 61)(18 37 22 33)(19 59 23 63)(20 39 24 35)(26 51 30 55)(28 53 32 49)(34 48 38 44)(36 42 40 46)(41 60 45 64)(43 62 47 58)
(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 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)
G:=sub<Sym(64)| (1,41,28,24)(2,42,29,17)(3,43,30,18)(4,44,31,19)(5,45,32,20)(6,46,25,21)(7,47,26,22)(8,48,27,23)(9,33,55,58)(10,34,56,59)(11,35,49,60)(12,36,50,61)(13,37,51,62)(14,38,52,63)(15,39,53,64)(16,40,54,57), (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,53,13,49)(10,54,14,50)(11,55,15,51)(12,56,16,52)(17,48,21,44)(18,41,22,45)(19,42,23,46)(20,43,24,47)(33,64,37,60)(34,57,38,61)(35,58,39,62)(36,59,40,63), (1,15,5,11)(2,50,6,54)(3,9,7,13)(4,52,8,56)(10,31,14,27)(12,25,16,29)(17,57,21,61)(18,37,22,33)(19,59,23,63)(20,39,24,35)(26,51,30,55)(28,53,32,49)(34,48,38,44)(36,42,40,46)(41,60,45,64)(43,62,47,58), (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), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59)>;
G:=Group( (1,41,28,24)(2,42,29,17)(3,43,30,18)(4,44,31,19)(5,45,32,20)(6,46,25,21)(7,47,26,22)(8,48,27,23)(9,33,55,58)(10,34,56,59)(11,35,49,60)(12,36,50,61)(13,37,51,62)(14,38,52,63)(15,39,53,64)(16,40,54,57), (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,53,13,49)(10,54,14,50)(11,55,15,51)(12,56,16,52)(17,48,21,44)(18,41,22,45)(19,42,23,46)(20,43,24,47)(33,64,37,60)(34,57,38,61)(35,58,39,62)(36,59,40,63), (1,15,5,11)(2,50,6,54)(3,9,7,13)(4,52,8,56)(10,31,14,27)(12,25,16,29)(17,57,21,61)(18,37,22,33)(19,59,23,63)(20,39,24,35)(26,51,30,55)(28,53,32,49)(34,48,38,44)(36,42,40,46)(41,60,45,64)(43,62,47,58), (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), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59) );
G=PermutationGroup([[(1,41,28,24),(2,42,29,17),(3,43,30,18),(4,44,31,19),(5,45,32,20),(6,46,25,21),(7,47,26,22),(8,48,27,23),(9,33,55,58),(10,34,56,59),(11,35,49,60),(12,36,50,61),(13,37,51,62),(14,38,52,63),(15,39,53,64),(16,40,54,57)], [(1,26,5,30),(2,27,6,31),(3,28,7,32),(4,29,8,25),(9,53,13,49),(10,54,14,50),(11,55,15,51),(12,56,16,52),(17,48,21,44),(18,41,22,45),(19,42,23,46),(20,43,24,47),(33,64,37,60),(34,57,38,61),(35,58,39,62),(36,59,40,63)], [(1,15,5,11),(2,50,6,54),(3,9,7,13),(4,52,8,56),(10,31,14,27),(12,25,16,29),(17,57,21,61),(18,37,22,33),(19,59,23,63),(20,39,24,35),(26,51,30,55),(28,53,32,49),(34,48,38,44),(36,42,40,46),(41,60,45,64),(43,62,47,58)], [(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)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(41,60),(42,61),(43,62),(44,63),(45,64),(46,57),(47,58),(48,59)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H | 8I | ··· | 8T | 8U | ··· | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C8○D4 |
| kernel | C42.681C23 | C2×C4×C8 | (C22×C8)⋊C2 | C4⋊M4(2) | C8×D4 | C8⋊6D4 | C22.26C24 | C2×C8○D4 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | C2×C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 1 | 2 | 8 | 4 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C42.681C23 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 2 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
| 16 | 15 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,16,0,0,2,16,0,0,0,0,0,16,0,0,1,0],[2,0,0,0,0,2,0,0,0,0,0,8,0,0,8,0],[16,0,0,0,15,1,0,0,0,0,0,1,0,0,1,0] >;
C42.681C23 in GAP, Magma, Sage, TeX
C_4^2._{681}C_2^3 % in TeX
G:=Group("C4^2.681C2^3"); // GroupNames label
G:=SmallGroup(128,1663);
// by ID
G=gap.SmallGroup(128,1663);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,1018,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*b^2*c,e*c*e=b^2*c,d*e=e*d>;
// generators/relations