p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.316D4, C42.614C23, C4.12C4≀C2, Q8⋊C8⋊34C2, C4⋊Q8.11C4, C4.2(C8○D4), C4.53(C2×Q16), (C2×C4).33Q16, C22⋊Q8.5C4, C4.96(C2×SD16), (C2×C4).68SD16, (C4×Q8).5C22, C4⋊C8.198C22, (C4×C8).364C22, C42.258(C2×C4), (C22×C4).573D4, C4.16(Q8⋊C4), C4⋊M4(2).13C2, C23.98(C22⋊C4), (C2×C42).1036C22, C22.11(Q8⋊C4), C23.37C23.3C2, (C2×C4×C8).10C2, C2.13(C2×C4≀C2), C4⋊C4.54(C2×C4), (C2×Q8).47(C2×C4), C2.6(C2×Q8⋊C4), (C2×C4).1142(C2×D4), (C2×C4).319(C22×C4), (C22×C4).397(C2×C4), (C2×C4).168(C22⋊C4), C22.169(C2×C22⋊C4), C2.19((C22×C8)⋊C2), SmallGroup(128,225)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.316D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 196 in 116 conjugacy classes, 56 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×Q8, C2×Q8, C4×C8, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), Q8⋊C8, C2×C4×C8, C4⋊M4(2), C23.37C23, C42.316D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C4≀C2, C2×C22⋊C4, C8○D4, C2×SD16, C2×Q16, (C22×C8)⋊C2, C2×Q8⋊C4, C2×C4≀C2, C42.316D4
►On 64 points
(1 51 59 34)(2 52 60 35)(3 53 61 36)(4 54 62 37)(5 55 63 38)(6 56 64 39)(7 49 57 40)(8 50 58 33)(9 45 30 24)(10 46 31 17)(11 47 32 18)(12 48 25 19)(13 41 26 20)(14 42 27 21)(15 43 28 22)(16 44 29 23)
(1 36 63 49)(2 37 64 50)(3 38 57 51)(4 39 58 52)(5 40 59 53)(6 33 60 54)(7 34 61 55)(8 35 62 56)(9 43 26 18)(10 44 27 19)(11 45 28 20)(12 46 29 21)(13 47 30 22)(14 48 31 23)(15 41 32 24)(16 42 25 17)
(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 19 36 10 63 44 49 27)(2 30 37 22 64 13 50 47)(3 42 38 25 57 17 51 16)(4 11 39 45 58 28 52 20)(5 23 40 14 59 48 53 31)(6 26 33 18 60 9 54 43)(7 46 34 29 61 21 55 12)(8 15 35 41 62 32 56 24)
G:=sub<Sym(64)| (1,51,59,34)(2,52,60,35)(3,53,61,36)(4,54,62,37)(5,55,63,38)(6,56,64,39)(7,49,57,40)(8,50,58,33)(9,45,30,24)(10,46,31,17)(11,47,32,18)(12,48,25,19)(13,41,26,20)(14,42,27,21)(15,43,28,22)(16,44,29,23), (1,36,63,49)(2,37,64,50)(3,38,57,51)(4,39,58,52)(5,40,59,53)(6,33,60,54)(7,34,61,55)(8,35,62,56)(9,43,26,18)(10,44,27,19)(11,45,28,20)(12,46,29,21)(13,47,30,22)(14,48,31,23)(15,41,32,24)(16,42,25,17), (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), (1,19,36,10,63,44,49,27)(2,30,37,22,64,13,50,47)(3,42,38,25,57,17,51,16)(4,11,39,45,58,28,52,20)(5,23,40,14,59,48,53,31)(6,26,33,18,60,9,54,43)(7,46,34,29,61,21,55,12)(8,15,35,41,62,32,56,24)>;
G:=Group( (1,51,59,34)(2,52,60,35)(3,53,61,36)(4,54,62,37)(5,55,63,38)(6,56,64,39)(7,49,57,40)(8,50,58,33)(9,45,30,24)(10,46,31,17)(11,47,32,18)(12,48,25,19)(13,41,26,20)(14,42,27,21)(15,43,28,22)(16,44,29,23), (1,36,63,49)(2,37,64,50)(3,38,57,51)(4,39,58,52)(5,40,59,53)(6,33,60,54)(7,34,61,55)(8,35,62,56)(9,43,26,18)(10,44,27,19)(11,45,28,20)(12,46,29,21)(13,47,30,22)(14,48,31,23)(15,41,32,24)(16,42,25,17), (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), (1,19,36,10,63,44,49,27)(2,30,37,22,64,13,50,47)(3,42,38,25,57,17,51,16)(4,11,39,45,58,28,52,20)(5,23,40,14,59,48,53,31)(6,26,33,18,60,9,54,43)(7,46,34,29,61,21,55,12)(8,15,35,41,62,32,56,24) );
G=PermutationGroup([[(1,51,59,34),(2,52,60,35),(3,53,61,36),(4,54,62,37),(5,55,63,38),(6,56,64,39),(7,49,57,40),(8,50,58,33),(9,45,30,24),(10,46,31,17),(11,47,32,18),(12,48,25,19),(13,41,26,20),(14,42,27,21),(15,43,28,22),(16,44,29,23)], [(1,36,63,49),(2,37,64,50),(3,38,57,51),(4,39,58,52),(5,40,59,53),(6,33,60,54),(7,34,61,55),(8,35,62,56),(9,43,26,18),(10,44,27,19),(11,45,28,20),(12,46,29,21),(13,47,30,22),(14,48,31,23),(15,41,32,24),(16,42,25,17)], [(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)], [(1,19,36,10,63,44,49,27),(2,30,37,22,64,13,50,47),(3,42,38,25,57,17,51,16),(4,11,39,45,58,28,52,20),(5,23,40,14,59,48,53,31),(6,26,33,18,60,9,54,43),(7,46,34,29,61,21,55,12),(8,15,35,41,62,32,56,24)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| 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 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C4≀C2 | C8○D4 |
| kernel | C42.316D4 | Q8⋊C8 | C2×C4×C8 | C4⋊M4(2) | C23.37C23 | C22⋊Q8 | C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C42.316D4 ►in GL4(𝔽17) generated by
| 1 | 2 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 11 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 11 | 3 | 0 | 0 |
| 16 | 6 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [1,16,0,0,2,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[0,3,0,0,11,6,0,0,0,0,9,0,0,0,0,9],[11,16,0,0,3,6,0,0,0,0,0,9,0,0,9,0] >;
C42.316D4 in GAP, Magma, Sage, TeX
C_4^2._{316}D_4 % in TeX
G:=Group("C4^2.316D4"); // GroupNames label
G:=SmallGroup(128,225);
// by ID
G=gap.SmallGroup(128,225);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,723,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations