p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.388D4, C4⋊C8.8C4, C4.31C4≀C2, C8⋊C4.9C4, (C2×C4).11C42, C42.37(C2×C4), C4.8(C8.C4), (C22×C4).23Q8, C23.15(C4⋊C4), (C22×C4).182D4, (C2×M4(2)).2C4, C4.19(C4.D4), (C4×M4(2)).11C2, C4⋊M4(2).6C2, C2.11(C42⋊6C4), C4.19(C4.10D4), C42.12C4.9C2, C2.7(C4.C42), (C2×C42).134C22, C2.8(C22.C42), C22.49(C2.C42), (C2×C4).72(C4⋊C4), (C22×C4).154(C2×C4), (C2×C4).302(C22⋊C4), SmallGroup(128,31)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.388D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=ab-1c3 >
Subgroups: 128 in 75 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×M4(2), C2×M4(2), C4×M4(2), C4⋊M4(2), C42.12C4, C42.388D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C4.D4, C4.10D4, C4≀C2, C8.C4, C42⋊6C4, C4.C42, C22.C42, C42.388D4
►On 64 points
(1 56 59 18)(2 49 60 19)(3 50 61 20)(4 51 62 21)(5 52 63 22)(6 53 64 23)(7 54 57 24)(8 55 58 17)(9 41 26 35)(10 42 27 36)(11 43 28 37)(12 44 29 38)(13 45 30 39)(14 46 31 40)(15 47 32 33)(16 48 25 34)
(1 7 5 3)(2 62 6 58)(4 64 8 60)(9 11 13 15)(10 25 14 29)(12 27 16 31)(17 49 21 53)(18 24 22 20)(19 51 23 55)(26 28 30 32)(33 35 37 39)(34 46 38 42)(36 48 40 44)(41 43 45 47)(50 56 54 52)(57 63 61 59)
(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 14 56 46 59 31 18 40)(2 37 49 11 60 43 19 28)(3 12 50 44 61 29 20 38)(4 35 51 9 62 41 21 26)(5 10 52 42 63 27 22 36)(6 33 53 15 64 47 23 32)(7 16 54 48 57 25 24 34)(8 39 55 13 58 45 17 30)
G:=sub<Sym(64)| (1,56,59,18)(2,49,60,19)(3,50,61,20)(4,51,62,21)(5,52,63,22)(6,53,64,23)(7,54,57,24)(8,55,58,17)(9,41,26,35)(10,42,27,36)(11,43,28,37)(12,44,29,38)(13,45,30,39)(14,46,31,40)(15,47,32,33)(16,48,25,34), (1,7,5,3)(2,62,6,58)(4,64,8,60)(9,11,13,15)(10,25,14,29)(12,27,16,31)(17,49,21,53)(18,24,22,20)(19,51,23,55)(26,28,30,32)(33,35,37,39)(34,46,38,42)(36,48,40,44)(41,43,45,47)(50,56,54,52)(57,63,61,59), (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), (1,14,56,46,59,31,18,40)(2,37,49,11,60,43,19,28)(3,12,50,44,61,29,20,38)(4,35,51,9,62,41,21,26)(5,10,52,42,63,27,22,36)(6,33,53,15,64,47,23,32)(7,16,54,48,57,25,24,34)(8,39,55,13,58,45,17,30)>;
G:=Group( (1,56,59,18)(2,49,60,19)(3,50,61,20)(4,51,62,21)(5,52,63,22)(6,53,64,23)(7,54,57,24)(8,55,58,17)(9,41,26,35)(10,42,27,36)(11,43,28,37)(12,44,29,38)(13,45,30,39)(14,46,31,40)(15,47,32,33)(16,48,25,34), (1,7,5,3)(2,62,6,58)(4,64,8,60)(9,11,13,15)(10,25,14,29)(12,27,16,31)(17,49,21,53)(18,24,22,20)(19,51,23,55)(26,28,30,32)(33,35,37,39)(34,46,38,42)(36,48,40,44)(41,43,45,47)(50,56,54,52)(57,63,61,59), (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), (1,14,56,46,59,31,18,40)(2,37,49,11,60,43,19,28)(3,12,50,44,61,29,20,38)(4,35,51,9,62,41,21,26)(5,10,52,42,63,27,22,36)(6,33,53,15,64,47,23,32)(7,16,54,48,57,25,24,34)(8,39,55,13,58,45,17,30) );
G=PermutationGroup([[(1,56,59,18),(2,49,60,19),(3,50,61,20),(4,51,62,21),(5,52,63,22),(6,53,64,23),(7,54,57,24),(8,55,58,17),(9,41,26,35),(10,42,27,36),(11,43,28,37),(12,44,29,38),(13,45,30,39),(14,46,31,40),(15,47,32,33),(16,48,25,34)], [(1,7,5,3),(2,62,6,58),(4,64,8,60),(9,11,13,15),(10,25,14,29),(12,27,16,31),(17,49,21,53),(18,24,22,20),(19,51,23,55),(26,28,30,32),(33,35,37,39),(34,46,38,42),(36,48,40,44),(41,43,45,47),(50,56,54,52),(57,63,61,59)], [(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)], [(1,14,56,46,59,31,18,40),(2,37,49,11,60,43,19,28),(3,12,50,44,61,29,20,38),(4,35,51,9,62,41,21,26),(5,10,52,42,63,27,22,36),(6,33,53,15,64,47,23,32),(7,16,54,48,57,25,24,34),(8,39,55,13,58,45,17,30)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | Q8 | C4≀C2 | C8.C4 | C4.D4 | C4.10D4 |
| kernel | C42.388D4 | C4×M4(2) | C4⋊M4(2) | C42.12C4 | C8⋊C4 | C4⋊C8 | C2×M4(2) | C42 | C22×C4 | C22×C4 | C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 8 | 8 | 1 | 1 |
Matrix representation of C42.388D4 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 14 | 2 |
| 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 8 | 16 |
| 0 | 0 | 9 | 9 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,16,0,0,0,0,0,8,14,0,0,0,2],[0,13,0,0,1,0,0,0,0,0,8,9,0,0,16,9] >;
C42.388D4 in GAP, Magma, Sage, TeX
C_4^2._{388}D_4 % in TeX
G:=Group("C4^2.388D4"); // GroupNames label
G:=SmallGroup(128,31);
// by ID
G=gap.SmallGroup(128,31);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,136,3924,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a*b^-1*c^3>;
// generators/relations