p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.84(C4×D4), C4⋊C4.313D4, (C2×C4)⋊15SD16, (C2×SD16)⋊15C4, C4.4(C4⋊D4), Q8⋊1(C22⋊C4), (C2×Q8).163D4, C2.11(C4×SD16), C2.7(D4⋊D4), C2.5(Q8⋊D4), C2.2(C4⋊SD16), C23.771(C2×D4), (C22×C4).690D4, C22.154(C4×D4), C22.4Q16⋊41C2, C22.96C22≀C2, C22.58(C4○D8), (C22×C8).38C22, C2.3(Q8.D4), (C22×SD16).7C2, C22.59(C2×SD16), C22.78(C8⋊C22), (C2×C42).280C22, (C22×D4).17C22, C2.11(SD16⋊C4), C22.117(C4⋊D4), (C22×C4).1367C23, C22.7C42⋊28C2, C4.66(C22.D4), C22.67(C8.C22), C24.3C22.5C2, (C22×Q8).388C22, C2.19(C23.23D4), (C2×C4×Q8)⋊1C2, (C2×C8).109(C2×C4), (C2×Q8⋊C4)⋊6C2, (C2×D4).73(C2×C4), (C2×C4).999(C2×D4), C4.14(C2×C22⋊C4), (C2×D4⋊C4).4C2, (C2×Q8).146(C2×C4), (C2×C4).564(C4○D4), (C2×C4⋊C4).768C22, (C2×C4).385(C22×C4), SmallGroup(128,612)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×SD16)⋊15C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=ab-1, dcd-1=ab6c >
Subgroups: 404 in 187 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, D4⋊C4, Q8⋊C4, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, C22.7C42, C22.4Q16, C24.3C22, C2×D4⋊C4, C2×Q8⋊C4, C2×C4×Q8, C22×SD16, (C2×SD16)⋊15C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.23D4, C4×SD16, SD16⋊C4, Q8⋊D4, D4⋊D4, C4⋊SD16, Q8.D4, (C2×SD16)⋊15C4
►On 64 points
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)
(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)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 34)(10 37)(11 40)(12 35)(13 38)(14 33)(15 36)(16 39)(25 48)(26 43)(27 46)(28 41)(29 44)(30 47)(31 42)(32 45)(49 60)(50 63)(51 58)(52 61)(53 64)(54 59)(55 62)(56 57)
(1 30 17 54)(2 62 18 42)(3 28 19 52)(4 60 20 48)(5 26 21 50)(6 58 22 46)(7 32 23 56)(8 64 24 44)(9 27 34 51)(10 59 35 47)(11 25 36 49)(12 57 37 45)(13 31 38 55)(14 63 39 43)(15 29 40 53)(16 61 33 41)
G:=sub<Sym(64)| (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(41,52)(42,53)(43,54)(44,55)(45,56)(46,49)(47,50)(48,51), (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)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45)(49,60)(50,63)(51,58)(52,61)(53,64)(54,59)(55,62)(56,57), (1,30,17,54)(2,62,18,42)(3,28,19,52)(4,60,20,48)(5,26,21,50)(6,58,22,46)(7,32,23,56)(8,64,24,44)(9,27,34,51)(10,59,35,47)(11,25,36,49)(12,57,37,45)(13,31,38,55)(14,63,39,43)(15,29,40,53)(16,61,33,41)>;
G:=Group( (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(41,52)(42,53)(43,54)(44,55)(45,56)(46,49)(47,50)(48,51), (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)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45)(49,60)(50,63)(51,58)(52,61)(53,64)(54,59)(55,62)(56,57), (1,30,17,54)(2,62,18,42)(3,28,19,52)(4,60,20,48)(5,26,21,50)(6,58,22,46)(7,32,23,56)(8,64,24,44)(9,27,34,51)(10,59,35,47)(11,25,36,49)(12,57,37,45)(13,31,38,55)(14,63,39,43)(15,29,40,53)(16,61,33,41) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(41,52),(42,53),(43,54),(44,55),(45,56),(46,49),(47,50),(48,51)], [(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),(2,22),(3,17),(4,20),(5,23),(6,18),(7,21),(8,24),(9,34),(10,37),(11,40),(12,35),(13,38),(14,33),(15,36),(16,39),(25,48),(26,43),(27,46),(28,41),(29,44),(30,47),(31,42),(32,45),(49,60),(50,63),(51,58),(52,61),(53,64),(54,59),(55,62),(56,57)], [(1,30,17,54),(2,62,18,42),(3,28,19,52),(4,60,20,48),(5,26,21,50),(6,58,22,46),(7,32,23,56),(8,64,24,44),(9,27,34,51),(10,59,35,47),(11,25,36,49),(12,57,37,45),(13,31,38,55),(14,63,39,43),(15,29,40,53),(16,61,33,41)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | SD16 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | (C2×SD16)⋊15C4 | C22.7C42 | C22.4Q16 | C24.3C22 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4×Q8 | C22×SD16 | C2×SD16 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of (C2×SD16)⋊15C4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 8 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 8 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 14 | 3 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 8 | 15 |
| 0 | 0 | 0 | 6 | 9 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,1,8,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,16],[13,0,0,0,0,0,14,3,0,0,0,3,3,0,0,0,0,0,8,6,0,0,0,15,9] >;
(C2×SD16)⋊15C4 in GAP, Magma, Sage, TeX
(C_2\times {\rm SD}_{16})\rtimes_{15}C_4 % in TeX
G:=Group("(C2xSD16):15C4"); // GroupNames label
G:=SmallGroup(128,612);
// by ID
G=gap.SmallGroup(128,612);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,352,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=a*b^-1,d*c*d^-1=a*b^6*c>;
// generators/relations