p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.71(C4×D4), C4⋊C4.215D4, (C2×C8).329D4, C4.10D4⋊3C4, M4(2).7(C2×C4), C2.3(D4.5D4), C2.4(D4.3D4), C4.28(C4.4D4), C4.C42.2C2, C23.265(C4○D4), C8○2M4(2).18C2, M4(2)⋊C4.2C2, (C22×C4).693C23, (C22×C8).395C22, C23.38D4.5C2, (C22×Q8).29C22, C22.128(C4⋊D4), C22.C42.11C2, C22.6(C42⋊2C2), C22.9(C42⋊C2), C4.90(C22.D4), C42⋊C2.273C22, (C2×M4(2)).198C22, C2.15(C24.C22), (C2×Q8).82(C2×C4), (C2×C4).1338(C2×D4), (C2×C4⋊C4).68C22, (C2×C4).15(C22×C4), (C2×C4).334(C4○D4), (C2×Q8⋊C4).32C2, (C2×C4.10D4).2C2, SmallGroup(128,662)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.10D4⋊3C4
G = < a,b,c,d | a4=d4=1, b4=a2, c2=bab-1=dad-1=a-1, dcd-1=ac=ca, cbc-1=a-1b3, dbd-1=ab3 >
Subgroups: 196 in 106 conjugacy classes, 48 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C4.10D4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C22×Q8, C4.C42, C22.C42, C8○2M4(2), C2×C4.10D4, C2×Q8⋊C4, C23.38D4, M4(2)⋊C4, C4.10D4⋊3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C24.C22, D4.3D4, D4.5D4, C4.10D4⋊3C4
►On 64 points
(1 23 5 19)(2 20 6 24)(3 17 7 21)(4 22 8 18)(9 62 13 58)(10 59 14 63)(11 64 15 60)(12 61 16 57)(25 40 29 36)(26 37 30 33)(27 34 31 38)(28 39 32 35)(41 52 45 56)(42 49 46 53)(43 54 47 50)(44 51 48 55)
(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 63 19 14 5 59 23 10)(2 13 24 62 6 9 20 58)(3 61 21 12 7 57 17 16)(4 11 18 60 8 15 22 64)(25 53 36 46 29 49 40 42)(26 45 33 52 30 41 37 56)(27 51 38 44 31 55 34 48)(28 43 35 50 32 47 39 54)
(1 51 21 46)(2 43 22 56)(3 49 23 44)(4 41 24 54)(5 55 17 42)(6 47 18 52)(7 53 19 48)(8 45 20 50)(9 32 60 33)(10 38 61 29)(11 30 62 39)(12 36 63 27)(13 28 64 37)(14 34 57 25)(15 26 58 35)(16 40 59 31)
G:=sub<Sym(64)| (1,23,5,19)(2,20,6,24)(3,17,7,21)(4,22,8,18)(9,62,13,58)(10,59,14,63)(11,64,15,60)(12,61,16,57)(25,40,29,36)(26,37,30,33)(27,34,31,38)(28,39,32,35)(41,52,45,56)(42,49,46,53)(43,54,47,50)(44,51,48,55), (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), (1,63,19,14,5,59,23,10)(2,13,24,62,6,9,20,58)(3,61,21,12,7,57,17,16)(4,11,18,60,8,15,22,64)(25,53,36,46,29,49,40,42)(26,45,33,52,30,41,37,56)(27,51,38,44,31,55,34,48)(28,43,35,50,32,47,39,54), (1,51,21,46)(2,43,22,56)(3,49,23,44)(4,41,24,54)(5,55,17,42)(6,47,18,52)(7,53,19,48)(8,45,20,50)(9,32,60,33)(10,38,61,29)(11,30,62,39)(12,36,63,27)(13,28,64,37)(14,34,57,25)(15,26,58,35)(16,40,59,31)>;
G:=Group( (1,23,5,19)(2,20,6,24)(3,17,7,21)(4,22,8,18)(9,62,13,58)(10,59,14,63)(11,64,15,60)(12,61,16,57)(25,40,29,36)(26,37,30,33)(27,34,31,38)(28,39,32,35)(41,52,45,56)(42,49,46,53)(43,54,47,50)(44,51,48,55), (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), (1,63,19,14,5,59,23,10)(2,13,24,62,6,9,20,58)(3,61,21,12,7,57,17,16)(4,11,18,60,8,15,22,64)(25,53,36,46,29,49,40,42)(26,45,33,52,30,41,37,56)(27,51,38,44,31,55,34,48)(28,43,35,50,32,47,39,54), (1,51,21,46)(2,43,22,56)(3,49,23,44)(4,41,24,54)(5,55,17,42)(6,47,18,52)(7,53,19,48)(8,45,20,50)(9,32,60,33)(10,38,61,29)(11,30,62,39)(12,36,63,27)(13,28,64,37)(14,34,57,25)(15,26,58,35)(16,40,59,31) );
G=PermutationGroup([[(1,23,5,19),(2,20,6,24),(3,17,7,21),(4,22,8,18),(9,62,13,58),(10,59,14,63),(11,64,15,60),(12,61,16,57),(25,40,29,36),(26,37,30,33),(27,34,31,38),(28,39,32,35),(41,52,45,56),(42,49,46,53),(43,54,47,50),(44,51,48,55)], [(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)], [(1,63,19,14,5,59,23,10),(2,13,24,62,6,9,20,58),(3,61,21,12,7,57,17,16),(4,11,18,60,8,15,22,64),(25,53,36,46,29,49,40,42),(26,45,33,52,30,41,37,56),(27,51,38,44,31,55,34,48),(28,43,35,50,32,47,39,54)], [(1,51,21,46),(2,43,22,56),(3,49,23,44),(4,41,24,54),(5,55,17,42),(6,47,18,52),(7,53,19,48),(8,45,20,50),(9,32,60,33),(10,38,61,29),(11,30,62,39),(12,36,63,27),(13,28,64,37),(14,34,57,25),(15,26,58,35),(16,40,59,31)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 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 | C4 | D4 | D4 | C4○D4 | C4○D4 | D4.3D4 | D4.5D4 |
| kernel | C4.10D4⋊3C4 | C4.C42 | C22.C42 | C8○2M4(2) | C2×C4.10D4 | C2×Q8⋊C4 | C23.38D4 | M4(2)⋊C4 | C4.10D4 | C4⋊C4 | C2×C8 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 6 | 2 | 2 | 2 |
Matrix representation of C4.10D4⋊3C4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 15 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 10 | 5 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 15 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 15 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,0,0,0,0,1,15,0,0,0,0,1,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,10,7,0,0,0,0,5,7,0,0,0,10,0,0,0,0,5,0,0,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,1,15,0,0,0,0,0,16,0,0,0,0,0,0,1,15,0,0,0,0,0,16] >;
C4.10D4⋊3C4 in GAP, Magma, Sage, TeX
C_4._{10}D_4\rtimes_3C_4 % in TeX
G:=Group("C4.10D4:3C4"); // GroupNames label
G:=SmallGroup(128,662);
// by ID
G=gap.SmallGroup(128,662);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2019,248,718]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^4=1,b^4=a^2,c^2=b*a*b^-1=d*a*d^-1=a^-1,d*c*d^-1=a*c=c*a,c*b*c^-1=a^-1*b^3,d*b*d^-1=a*b^3>;
// generators/relations