p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.294C23, (C8×D4)⋊43C2, (C8×Q8)⋊31C2, C8⋊6D4⋊37C2, C8⋊9D4⋊37C2, C8⋊4Q8⋊37C2, C4.39(C8○D4), C8.87(C4○D4), C4⋊D4.23C4, C22⋊Q8.23C4, C4⋊C8.364C22, (C4×M4(2))⋊37C2, (C2×C8).614C23, C42.217(C2×C4), (C2×C4).666C24, C42⋊2C2.3C4, (C4×C8).335C22, C4.4D4.19C4, C42.C2.19C4, C8○2M4(2)⋊36C2, (C4×D4).295C22, C23.40(C22×C4), (C4×Q8).280C22, C22.D4.7C4, C42.12C4⋊51C2, C8⋊C4.175C22, C22⋊C8.142C22, C2.24(Q8○M4(2)), (C2×C42).776C22, (C22×C4).936C23, (C22×C8).448C22, C22.191(C23×C4), C42.7C22⋊25C2, C42⋊C2.309C22, (C2×M4(2)).368C22, C23.36C23.15C2, C8⋊C4○(C4⋊C8), C2.48(C4×C4○D4), C2.26(C2×C8○D4), C4⋊C4.166(C2×C4), C4.317(C2×C4○D4), (C2×D4).142(C2×C4), C22⋊C4.18(C2×C4), (C2×Q8).165(C2×C4), (C2×C4).273(C22×C4), (C22×C4).350(C2×C4), SmallGroup(128,1701)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.294C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b, e2=a2, ab=ba, cac-1=a-1b2, dad-1=eae-1=ab2, bc=cb, bd=db, be=eb, dcd-1=a2c, ce=ec, de=ed >
Subgroups: 252 in 185 conjugacy classes, 130 normal (52 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C4×C8, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×M4(2), C2×M4(2), C4×M4(2), C8○2M4(2), C42.12C4, C42.7C22, C8×D4, C8⋊9D4, C8⋊6D4, C8⋊6D4, C8×Q8, C8⋊4Q8, C23.36C23, C42.294C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C8○D4, C23×C4, C2×C4○D4, C4×C4○D4, C2×C8○D4, Q8○M4(2), C42.294C23
►On 64 points
(1 37 22 48)(2 34 23 45)(3 39 24 42)(4 36 17 47)(5 33 18 44)(6 38 19 41)(7 35 20 46)(8 40 21 43)(9 55 32 62)(10 52 25 59)(11 49 26 64)(12 54 27 61)(13 51 28 58)(14 56 29 63)(15 53 30 60)(16 50 31 57)
(1 24 5 20)(2 17 6 21)(3 18 7 22)(4 19 8 23)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)(49 58 53 62)(50 59 54 63)(51 60 55 64)(52 61 56 57)
(1 48 5 44)(2 38 6 34)(3 42 7 46)(4 40 8 36)(9 58 13 62)(10 52 14 56)(11 60 15 64)(12 54 16 50)(17 43 21 47)(18 33 22 37)(19 45 23 41)(20 35 24 39)(25 59 29 63)(26 53 30 49)(27 61 31 57)(28 55 32 51)
(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 61 22 54)(2 62 23 55)(3 63 24 56)(4 64 17 49)(5 57 18 50)(6 58 19 51)(7 59 20 52)(8 60 21 53)(9 41 32 38)(10 42 25 39)(11 43 26 40)(12 44 27 33)(13 45 28 34)(14 46 29 35)(15 47 30 36)(16 48 31 37)
G:=sub<Sym(64)| (1,37,22,48)(2,34,23,45)(3,39,24,42)(4,36,17,47)(5,33,18,44)(6,38,19,41)(7,35,20,46)(8,40,21,43)(9,55,32,62)(10,52,25,59)(11,49,26,64)(12,54,27,61)(13,51,28,58)(14,56,29,63)(15,53,30,60)(16,50,31,57), (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45)(49,58,53,62)(50,59,54,63)(51,60,55,64)(52,61,56,57), (1,48,5,44)(2,38,6,34)(3,42,7,46)(4,40,8,36)(9,58,13,62)(10,52,14,56)(11,60,15,64)(12,54,16,50)(17,43,21,47)(18,33,22,37)(19,45,23,41)(20,35,24,39)(25,59,29,63)(26,53,30,49)(27,61,31,57)(28,55,32,51), (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), (1,61,22,54)(2,62,23,55)(3,63,24,56)(4,64,17,49)(5,57,18,50)(6,58,19,51)(7,59,20,52)(8,60,21,53)(9,41,32,38)(10,42,25,39)(11,43,26,40)(12,44,27,33)(13,45,28,34)(14,46,29,35)(15,47,30,36)(16,48,31,37)>;
G:=Group( (1,37,22,48)(2,34,23,45)(3,39,24,42)(4,36,17,47)(5,33,18,44)(6,38,19,41)(7,35,20,46)(8,40,21,43)(9,55,32,62)(10,52,25,59)(11,49,26,64)(12,54,27,61)(13,51,28,58)(14,56,29,63)(15,53,30,60)(16,50,31,57), (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45)(49,58,53,62)(50,59,54,63)(51,60,55,64)(52,61,56,57), (1,48,5,44)(2,38,6,34)(3,42,7,46)(4,40,8,36)(9,58,13,62)(10,52,14,56)(11,60,15,64)(12,54,16,50)(17,43,21,47)(18,33,22,37)(19,45,23,41)(20,35,24,39)(25,59,29,63)(26,53,30,49)(27,61,31,57)(28,55,32,51), (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), (1,61,22,54)(2,62,23,55)(3,63,24,56)(4,64,17,49)(5,57,18,50)(6,58,19,51)(7,59,20,52)(8,60,21,53)(9,41,32,38)(10,42,25,39)(11,43,26,40)(12,44,27,33)(13,45,28,34)(14,46,29,35)(15,47,30,36)(16,48,31,37) );
G=PermutationGroup([[(1,37,22,48),(2,34,23,45),(3,39,24,42),(4,36,17,47),(5,33,18,44),(6,38,19,41),(7,35,20,46),(8,40,21,43),(9,55,32,62),(10,52,25,59),(11,49,26,64),(12,54,27,61),(13,51,28,58),(14,56,29,63),(15,53,30,60),(16,50,31,57)], [(1,24,5,20),(2,17,6,21),(3,18,7,22),(4,19,8,23),(9,26,13,30),(10,27,14,31),(11,28,15,32),(12,29,16,25),(33,46,37,42),(34,47,38,43),(35,48,39,44),(36,41,40,45),(49,58,53,62),(50,59,54,63),(51,60,55,64),(52,61,56,57)], [(1,48,5,44),(2,38,6,34),(3,42,7,46),(4,40,8,36),(9,58,13,62),(10,52,14,56),(11,60,15,64),(12,54,16,50),(17,43,21,47),(18,33,22,37),(19,45,23,41),(20,35,24,39),(25,59,29,63),(26,53,30,49),(27,61,31,57),(28,55,32,51)], [(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)], [(1,61,22,54),(2,62,23,55),(3,63,24,56),(4,64,17,49),(5,57,18,50),(6,58,19,51),(7,59,20,52),(8,60,21,53),(9,41,32,38),(10,42,25,39),(11,43,26,40),(12,44,27,33),(13,45,28,34),(14,46,29,35),(15,47,30,36),(16,48,31,37)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4S | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | C42.294C23 | C4×M4(2) | C8○2M4(2) | C42.12C4 | C42.7C22 | C8×D4 | C8⋊9D4 | C8⋊6D4 | C8×Q8 | C8⋊4Q8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 2 |
Matrix representation of C42.294C23 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 0 | 4 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 15 | 4 |
| 0 | 0 | 0 | 2 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,4,4,0,0,9,13],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,16,0,0,0,0,0,13,0,0,0,8,4],[0,1,0,0,16,0,0,0,0,0,15,0,0,0,4,2],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,15,16] >;
C42.294C23 in GAP, Magma, Sage, TeX
C_4^2._{294}C_2^3 % in TeX
G:=Group("C4^2.294C2^3"); // GroupNames label
G:=SmallGroup(128,1701);
// by ID
G=gap.SmallGroup(128,1701);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,521,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=a^2*b,e^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=e*a*e^-1=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,c*e=e*c,d*e=e*d>;
// generators/relations