p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.19C23, C4⋊C4.340D4, C4○D4.19D4, D4.10(C2×D4), Q8.10(C2×D4), C2.9(Q8○D8), C4⋊2Q16⋊21C2, D4.D4⋊4C2, C4⋊C8.41C22, (C2×C8).21C23, C4.77(C22×D4), D4.2D4⋊17C2, C4⋊C4.387C23, (C2×C4).250C24, Q8.D4⋊17C2, C22⋊C4.141D4, (C4×D4).68C22, C23.447(C2×D4), C4⋊Q8.101C22, (C2×Q8).43C23, (C4×Q8).65C22, C4.171(C4⋊D4), C2.15(D4○SD16), (C2×D4).389C23, (C2×D8).118C22, C23.36D4⋊8C2, C22.7(C4⋊D4), (C2×Q16).54C22, (C2×SD16).7C22, D4⋊C4.22C22, C42.6C22⋊8C2, (C22×C4).980C23, (C22×C8).179C22, C4.4D4.27C22, C22.510(C22×D4), C23.33C23⋊8C2, C23.38C23⋊9C2, Q8⋊C4.170C22, (C2×M4(2)).57C22, (C22×Q8).276C22, C42⋊C2.105C22, (C2×C4○D8).6C2, C4.160(C2×C4○D4), (C2×C4).470(C2×D4), C2.68(C2×C4⋊D4), (C2×Q8⋊C4)⋊29C2, (C2×C8.C22)⋊17C2, (C2×C4).281(C4○D4), (C2×C4⋊C4).584C22, (C2×C4○D4).122C22, SmallGroup(128,1778)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.19C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=b2, e2=a2, ab=ba, cac-1=a-1, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >
Subgroups: 436 in 236 conjugacy classes, 100 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C8.C22, C22×Q8, C2×C4○D4, C2×Q8⋊C4, C23.36D4, C42.6C22, D4.D4, C4⋊2Q16, D4.2D4, Q8.D4, C23.33C23, C23.38C23, C2×C4○D8, C2×C8.C22, C42.19C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, D4○SD16, Q8○D8, C42.19C23
►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 57 12)(2 24 58 9)(3 21 59 10)(4 22 60 11)(5 47 40 27)(6 48 37 28)(7 45 38 25)(8 46 39 26)(13 36 19 53)(14 33 20 54)(15 34 17 55)(16 35 18 56)(29 61 44 52)(30 62 41 49)(31 63 42 50)(32 64 43 51)
(1 53 57 36)(2 56 58 35)(3 55 59 34)(4 54 60 33)(5 30 40 41)(6 29 37 44)(7 32 38 43)(8 31 39 42)(9 16 24 18)(10 15 21 17)(11 14 22 20)(12 13 23 19)(25 64 45 51)(26 63 46 50)(27 62 47 49)(28 61 48 52)
(1 37)(2 38)(3 39)(4 40)(5 60)(6 57)(7 58)(8 59)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)(33 49)(34 50)(35 51)(36 52)(53 61)(54 62)(55 63)(56 64)
(1 33 3 35)(2 55 4 53)(5 52 7 50)(6 62 8 64)(9 17 11 19)(10 16 12 14)(13 24 15 22)(18 23 20 21)(25 42 27 44)(26 32 28 30)(29 45 31 47)(34 60 36 58)(37 49 39 51)(38 63 40 61)(41 46 43 48)(54 59 56 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), (1,23,57,12)(2,24,58,9)(3,21,59,10)(4,22,60,11)(5,47,40,27)(6,48,37,28)(7,45,38,25)(8,46,39,26)(13,36,19,53)(14,33,20,54)(15,34,17,55)(16,35,18,56)(29,61,44,52)(30,62,41,49)(31,63,42,50)(32,64,43,51), (1,53,57,36)(2,56,58,35)(3,55,59,34)(4,54,60,33)(5,30,40,41)(6,29,37,44)(7,32,38,43)(8,31,39,42)(9,16,24,18)(10,15,21,17)(11,14,22,20)(12,13,23,19)(25,64,45,51)(26,63,46,50)(27,62,47,49)(28,61,48,52), (1,37)(2,38)(3,39)(4,40)(5,60)(6,57)(7,58)(8,59)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45)(33,49)(34,50)(35,51)(36,52)(53,61)(54,62)(55,63)(56,64), (1,33,3,35)(2,55,4,53)(5,52,7,50)(6,62,8,64)(9,17,11,19)(10,16,12,14)(13,24,15,22)(18,23,20,21)(25,42,27,44)(26,32,28,30)(29,45,31,47)(34,60,36,58)(37,49,39,51)(38,63,40,61)(41,46,43,48)(54,59,56,57)>;
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), (1,23,57,12)(2,24,58,9)(3,21,59,10)(4,22,60,11)(5,47,40,27)(6,48,37,28)(7,45,38,25)(8,46,39,26)(13,36,19,53)(14,33,20,54)(15,34,17,55)(16,35,18,56)(29,61,44,52)(30,62,41,49)(31,63,42,50)(32,64,43,51), (1,53,57,36)(2,56,58,35)(3,55,59,34)(4,54,60,33)(5,30,40,41)(6,29,37,44)(7,32,38,43)(8,31,39,42)(9,16,24,18)(10,15,21,17)(11,14,22,20)(12,13,23,19)(25,64,45,51)(26,63,46,50)(27,62,47,49)(28,61,48,52), (1,37)(2,38)(3,39)(4,40)(5,60)(6,57)(7,58)(8,59)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45)(33,49)(34,50)(35,51)(36,52)(53,61)(54,62)(55,63)(56,64), (1,33,3,35)(2,55,4,53)(5,52,7,50)(6,62,8,64)(9,17,11,19)(10,16,12,14)(13,24,15,22)(18,23,20,21)(25,42,27,44)(26,32,28,30)(29,45,31,47)(34,60,36,58)(37,49,39,51)(38,63,40,61)(41,46,43,48)(54,59,56,57) );
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)], [(1,23,57,12),(2,24,58,9),(3,21,59,10),(4,22,60,11),(5,47,40,27),(6,48,37,28),(7,45,38,25),(8,46,39,26),(13,36,19,53),(14,33,20,54),(15,34,17,55),(16,35,18,56),(29,61,44,52),(30,62,41,49),(31,63,42,50),(32,64,43,51)], [(1,53,57,36),(2,56,58,35),(3,55,59,34),(4,54,60,33),(5,30,40,41),(6,29,37,44),(7,32,38,43),(8,31,39,42),(9,16,24,18),(10,15,21,17),(11,14,22,20),(12,13,23,19),(25,64,45,51),(26,63,46,50),(27,62,47,49),(28,61,48,52)], [(1,37),(2,38),(3,39),(4,40),(5,60),(6,57),(7,58),(8,59),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,42),(18,43),(19,44),(20,41),(21,46),(22,47),(23,48),(24,45),(33,49),(34,50),(35,51),(36,52),(53,61),(54,62),(55,63),(56,64)], [(1,33,3,35),(2,55,4,53),(5,52,7,50),(6,62,8,64),(9,17,11,19),(10,16,12,14),(13,24,15,22),(18,23,20,21),(25,42,27,44),(26,32,28,30),(29,45,31,47),(34,60,36,58),(37,49,39,51),(38,63,40,61),(41,46,43,48),(54,59,56,57)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 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 | D4 | C4○D4 | D4○SD16 | Q8○D8 |
| kernel | C42.19C23 | C2×Q8⋊C4 | C23.36D4 | C42.6C22 | D4.D4 | C4⋊2Q16 | D4.2D4 | Q8.D4 | C23.33C23 | C23.38C23 | C2×C4○D8 | C2×C8.C22 | C22⋊C4 | C4⋊C4 | C4○D4 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.19C23 ►in GL6(𝔽17)
| 9 | 13 | 0 | 0 | 0 | 0 |
| 12 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 12 | 0 |
| 0 | 0 | 0 | 5 | 0 | 12 |
| 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 9 | 13 | 0 | 0 | 0 | 0 |
| 12 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 | 0 | 5 |
| 0 | 0 | 5 | 0 | 12 | 0 |
| 0 | 0 | 0 | 5 | 0 | 12 |
G:=sub<GL(6,GF(17))| [9,12,0,0,0,0,13,8,0,0,0,0,0,0,5,0,12,0,0,0,0,5,0,12,0,0,12,0,12,0,0,0,0,12,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,3,3,0,0,0,0,3,14],[9,12,0,0,0,0,13,8,0,0,0,0,0,0,5,0,5,0,0,0,0,5,0,5,0,0,5,0,12,0,0,0,0,5,0,12] >;
C42.19C23 in GAP, Magma, Sage, TeX
C_4^2._{19}C_2^3 % in TeX
G:=Group("C4^2.19C2^3"); // GroupNames label
G:=SmallGroup(128,1778);
// by ID
G=gap.SmallGroup(128,1778);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=b^2,e^2=a^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,d*e=e*d>;
// generators/relations