p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.21C23, (C2×Q8)⋊7Q8, Q8.5(C2×Q8), C4⋊C4.342D4, Q8⋊Q8⋊3C2, Q8.Q8⋊14C2, C4.Q16⋊20C2, C4⋊C8.45C22, C4⋊C4.45C23, (C2×C8).29C23, C2.13(Q8○D8), C4.33(C22×Q8), (C2×C4).280C24, C22⋊C4.143D4, C23.449(C2×D4), C4⋊Q8.102C22, C4.68(C22⋊Q8), (C4×Q8).66C22, C2.21(D4○SD16), (C2×Q8).368C23, C4.Q8.148C22, C2.D8.167C22, C42.C2.9C22, (C22×C8).182C22, (C22×C4).999C23, Q8⋊C4.27C22, C23.25D4.6C2, C23.38D4.3C2, C22.540(C22×D4), C22.11(C22⋊Q8), M4(2)⋊C4.11C2, (C2×M4(2)).69C22, C42.6C22.3C2, (C22×Q8).288C22, C42⋊C2.119C22, C23.32C23.5C2, C23.41C23.7C2, C4.90(C2×C4○D4), (C2×C4).482(C2×D4), (C2×C4).104(C2×Q8), C2.61(C2×C22⋊Q8), (C2×C4).482(C4○D4), (C2×C4⋊C4).606C22, (C2×Q8⋊C4).26C2, SmallGroup(128,1814)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.21C23
G = < a,b,c,d,e | a4=b4=1, c2=e2=a2b2, d2=b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
Subgroups: 292 in 181 conjugacy classes, 100 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C22×Q8, C2×Q8⋊C4, C23.38D4, C42.6C22, C23.25D4, M4(2)⋊C4, Q8⋊Q8, C4.Q16, Q8.Q8, C23.32C23, C23.41C23, C42.21C23
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D4○SD16, Q8○D8, C42.21C23
►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 29 12 38)(2 30 9 39)(3 31 10 40)(4 32 11 37)(5 42 23 59)(6 43 24 60)(7 44 21 57)(8 41 22 58)(13 52 25 35)(14 49 26 36)(15 50 27 33)(16 51 28 34)(17 54 62 45)(18 55 63 46)(19 56 64 47)(20 53 61 48)
(1 48 10 55)(2 56 11 45)(3 46 12 53)(4 54 9 47)(5 35 21 50)(6 51 22 36)(7 33 23 52)(8 49 24 34)(13 44 27 59)(14 60 28 41)(15 42 25 57)(16 58 26 43)(17 39 64 32)(18 29 61 40)(19 37 62 30)(20 31 63 38)
(1 13 12 25)(2 14 9 26)(3 15 10 27)(4 16 11 28)(5 55 23 46)(6 56 24 47)(7 53 21 48)(8 54 22 45)(17 58 62 41)(18 59 63 42)(19 60 64 43)(20 57 61 44)(29 35 38 52)(30 36 39 49)(31 33 40 50)(32 34 37 51)
(1 9 10 4)(2 3 11 12)(5 24 21 8)(6 7 22 23)(13 26 27 16)(14 15 28 25)(17 18 64 61)(19 20 62 63)(29 39 40 32)(30 31 37 38)(33 51 52 36)(34 35 49 50)(41 42 60 57)(43 44 58 59)(45 46 56 53)(47 48 54 55)
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), (1,29,12,38)(2,30,9,39)(3,31,10,40)(4,32,11,37)(5,42,23,59)(6,43,24,60)(7,44,21,57)(8,41,22,58)(13,52,25,35)(14,49,26,36)(15,50,27,33)(16,51,28,34)(17,54,62,45)(18,55,63,46)(19,56,64,47)(20,53,61,48), (1,48,10,55)(2,56,11,45)(3,46,12,53)(4,54,9,47)(5,35,21,50)(6,51,22,36)(7,33,23,52)(8,49,24,34)(13,44,27,59)(14,60,28,41)(15,42,25,57)(16,58,26,43)(17,39,64,32)(18,29,61,40)(19,37,62,30)(20,31,63,38), (1,13,12,25)(2,14,9,26)(3,15,10,27)(4,16,11,28)(5,55,23,46)(6,56,24,47)(7,53,21,48)(8,54,22,45)(17,58,62,41)(18,59,63,42)(19,60,64,43)(20,57,61,44)(29,35,38,52)(30,36,39,49)(31,33,40,50)(32,34,37,51), (1,9,10,4)(2,3,11,12)(5,24,21,8)(6,7,22,23)(13,26,27,16)(14,15,28,25)(17,18,64,61)(19,20,62,63)(29,39,40,32)(30,31,37,38)(33,51,52,36)(34,35,49,50)(41,42,60,57)(43,44,58,59)(45,46,56,53)(47,48,54,55)>;
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), (1,29,12,38)(2,30,9,39)(3,31,10,40)(4,32,11,37)(5,42,23,59)(6,43,24,60)(7,44,21,57)(8,41,22,58)(13,52,25,35)(14,49,26,36)(15,50,27,33)(16,51,28,34)(17,54,62,45)(18,55,63,46)(19,56,64,47)(20,53,61,48), (1,48,10,55)(2,56,11,45)(3,46,12,53)(4,54,9,47)(5,35,21,50)(6,51,22,36)(7,33,23,52)(8,49,24,34)(13,44,27,59)(14,60,28,41)(15,42,25,57)(16,58,26,43)(17,39,64,32)(18,29,61,40)(19,37,62,30)(20,31,63,38), (1,13,12,25)(2,14,9,26)(3,15,10,27)(4,16,11,28)(5,55,23,46)(6,56,24,47)(7,53,21,48)(8,54,22,45)(17,58,62,41)(18,59,63,42)(19,60,64,43)(20,57,61,44)(29,35,38,52)(30,36,39,49)(31,33,40,50)(32,34,37,51), (1,9,10,4)(2,3,11,12)(5,24,21,8)(6,7,22,23)(13,26,27,16)(14,15,28,25)(17,18,64,61)(19,20,62,63)(29,39,40,32)(30,31,37,38)(33,51,52,36)(34,35,49,50)(41,42,60,57)(43,44,58,59)(45,46,56,53)(47,48,54,55) );
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)], [(1,29,12,38),(2,30,9,39),(3,31,10,40),(4,32,11,37),(5,42,23,59),(6,43,24,60),(7,44,21,57),(8,41,22,58),(13,52,25,35),(14,49,26,36),(15,50,27,33),(16,51,28,34),(17,54,62,45),(18,55,63,46),(19,56,64,47),(20,53,61,48)], [(1,48,10,55),(2,56,11,45),(3,46,12,53),(4,54,9,47),(5,35,21,50),(6,51,22,36),(7,33,23,52),(8,49,24,34),(13,44,27,59),(14,60,28,41),(15,42,25,57),(16,58,26,43),(17,39,64,32),(18,29,61,40),(19,37,62,30),(20,31,63,38)], [(1,13,12,25),(2,14,9,26),(3,15,10,27),(4,16,11,28),(5,55,23,46),(6,56,24,47),(7,53,21,48),(8,54,22,45),(17,58,62,41),(18,59,63,42),(19,60,64,43),(20,57,61,44),(29,35,38,52),(30,36,39,49),(31,33,40,50),(32,34,37,51)], [(1,9,10,4),(2,3,11,12),(5,24,21,8),(6,7,22,23),(13,26,27,16),(14,15,28,25),(17,18,64,61),(19,20,62,63),(29,39,40,32),(30,31,37,38),(33,51,52,36),(34,35,49,50),(41,42,60,57),(43,44,58,59),(45,46,56,53),(47,48,54,55)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 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 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 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 | D4 | D4 | Q8 | C4○D4 | D4○SD16 | Q8○D8 |
| kernel | C42.21C23 | C2×Q8⋊C4 | C23.38D4 | C42.6C22 | C23.25D4 | M4(2)⋊C4 | Q8⋊Q8 | C4.Q16 | Q8.Q8 | C23.32C23 | C23.41C23 | C22⋊C4 | C4⋊C4 | C2×Q8 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.21C23 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 16 | 1 | 16 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 16 |
| 0 | 0 | 2 | 16 | 1 | 16 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 10 | 0 | 0 | 0 | 0 |
| 10 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 4 | 0 | 4 |
| 0 | 0 | 5 | 10 | 11 | 6 |
| 0 | 0 | 0 | 0 | 13 | 6 |
| 0 | 0 | 0 | 0 | 6 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 | 0 | 1 |
| 0 | 0 | 3 | 8 | 10 | 7 |
| 0 | 0 | 0 | 0 | 16 | 7 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 15 | 1 | 16 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,2,0,0,0,0,0,16,16,0,0,16,0,1,0,0,0,1,1,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,16,16,0,0,0,0,0,1,0,16,0,0,16,16,1,0],[1,10,0,0,0,0,10,16,0,0,0,0,0,0,7,5,0,0,0,0,4,10,0,0,0,0,0,11,13,6,0,0,4,6,6,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,9,3,0,0,0,0,1,8,0,0,0,0,0,10,16,7,0,0,1,7,7,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,15,0,0,0,0,0,1,1,0,0,0,0,16,0,0,0,0,1,1,0] >;
C42.21C23 in GAP, Magma, Sage, TeX
C_4^2._{21}C_2^3 % in TeX
G:=Group("C4^2.21C2^3"); // GroupNames label
G:=SmallGroup(128,1814);
// by ID
G=gap.SmallGroup(128,1814);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,352,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=e^2=a^2*b^2,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,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