p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4⋊C4.94D4, (C2×Q8).96D4, (C2×D4).106D4, C4.23(C4⋊D4), C23.914(C2×D4), (C22×C4).149D4, C2.33(D4⋊D4), (C22×C8).74C22, C22.224C22≀C2, C2.17(D4.2D4), C2.33(D4.7D4), C22.111(C4○D8), (C2×C42).365C22, C2.17(Q8.D4), (C22×D4).80C22, (C22×Q8).66C22, C22.231(C4⋊D4), C22.139(C8⋊C22), (C22×C4).1448C23, C23.65C23⋊6C2, C22.7C42⋊12C2, C22.91(C4.4D4), C4.72(C22.D4), C2.15(C23.10D4), C22.127(C8.C22), C2.8(C42.78C22), C2.8(C42.28C22), (C2×Q8⋊C4)⋊15C2, (C2×C4).1040(C2×D4), (C2×D4⋊C4).13C2, (C2×C4).879(C4○D4), (C2×C4⋊C4).127C22, (C2×C4.4D4).12C2, SmallGroup(128,774)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4.94D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b-1, dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 376 in 159 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4.4D4, C22×C8, C22×D4, C22×Q8, C22.7C42, C23.65C23, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.4D4, C4⋊C4.94D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C4○D8, C8⋊C22, C8.C22, C23.10D4, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, C42.78C22, C42.28C22, C4⋊C4.94D4
►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 63 11 58)(2 62 12 57)(3 61 9 60)(4 64 10 59)(5 15 21 52)(6 14 22 51)(7 13 23 50)(8 16 24 49)(17 34 39 53)(18 33 40 56)(19 36 37 55)(20 35 38 54)(25 44 30 45)(26 43 31 48)(27 42 32 47)(28 41 29 46)
(1 33 7 44)(2 34 8 41)(3 35 5 42)(4 36 6 43)(9 54 21 47)(10 55 22 48)(11 56 23 45)(12 53 24 46)(13 27 63 20)(14 28 64 17)(15 25 61 18)(16 26 62 19)(29 59 39 51)(30 60 40 52)(31 57 37 49)(32 58 38 50)
(1 42 3 44)(2 41 4 43)(5 33 7 35)(6 36 8 34)(9 45 11 47)(10 48 12 46)(13 39 15 37)(14 38 16 40)(17 52 19 50)(18 51 20 49)(21 56 23 54)(22 55 24 53)(25 59 27 57)(26 58 28 60)(29 61 31 63)(30 64 32 62)
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), (1,63,11,58)(2,62,12,57)(3,61,9,60)(4,64,10,59)(5,15,21,52)(6,14,22,51)(7,13,23,50)(8,16,24,49)(17,34,39,53)(18,33,40,56)(19,36,37,55)(20,35,38,54)(25,44,30,45)(26,43,31,48)(27,42,32,47)(28,41,29,46), (1,33,7,44)(2,34,8,41)(3,35,5,42)(4,36,6,43)(9,54,21,47)(10,55,22,48)(11,56,23,45)(12,53,24,46)(13,27,63,20)(14,28,64,17)(15,25,61,18)(16,26,62,19)(29,59,39,51)(30,60,40,52)(31,57,37,49)(32,58,38,50), (1,42,3,44)(2,41,4,43)(5,33,7,35)(6,36,8,34)(9,45,11,47)(10,48,12,46)(13,39,15,37)(14,38,16,40)(17,52,19,50)(18,51,20,49)(21,56,23,54)(22,55,24,53)(25,59,27,57)(26,58,28,60)(29,61,31,63)(30,64,32,62)>;
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), (1,63,11,58)(2,62,12,57)(3,61,9,60)(4,64,10,59)(5,15,21,52)(6,14,22,51)(7,13,23,50)(8,16,24,49)(17,34,39,53)(18,33,40,56)(19,36,37,55)(20,35,38,54)(25,44,30,45)(26,43,31,48)(27,42,32,47)(28,41,29,46), (1,33,7,44)(2,34,8,41)(3,35,5,42)(4,36,6,43)(9,54,21,47)(10,55,22,48)(11,56,23,45)(12,53,24,46)(13,27,63,20)(14,28,64,17)(15,25,61,18)(16,26,62,19)(29,59,39,51)(30,60,40,52)(31,57,37,49)(32,58,38,50), (1,42,3,44)(2,41,4,43)(5,33,7,35)(6,36,8,34)(9,45,11,47)(10,48,12,46)(13,39,15,37)(14,38,16,40)(17,52,19,50)(18,51,20,49)(21,56,23,54)(22,55,24,53)(25,59,27,57)(26,58,28,60)(29,61,31,63)(30,64,32,62) );
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)], [(1,63,11,58),(2,62,12,57),(3,61,9,60),(4,64,10,59),(5,15,21,52),(6,14,22,51),(7,13,23,50),(8,16,24,49),(17,34,39,53),(18,33,40,56),(19,36,37,55),(20,35,38,54),(25,44,30,45),(26,43,31,48),(27,42,32,47),(28,41,29,46)], [(1,33,7,44),(2,34,8,41),(3,35,5,42),(4,36,6,43),(9,54,21,47),(10,55,22,48),(11,56,23,45),(12,53,24,46),(13,27,63,20),(14,28,64,17),(15,25,61,18),(16,26,62,19),(29,59,39,51),(30,60,40,52),(31,57,37,49),(32,58,38,50)], [(1,42,3,44),(2,41,4,43),(5,33,7,35),(6,36,8,34),(9,45,11,47),(10,48,12,46),(13,39,15,37),(14,38,16,40),(17,52,19,50),(18,51,20,49),(21,56,23,54),(22,55,24,53),(25,59,27,57),(26,58,28,60),(29,61,31,63),(30,64,32,62)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C4⋊C4.94D4 | C22.7C42 | C23.65C23 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.4D4 | C4⋊C4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 6 | 8 | 1 | 1 |
Matrix representation of C4⋊C4.94D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 16 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,13,16,0,0,0,0,0,4,0,0,0,0,0,0,3,3,0,0,0,0,3,14],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,9,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0] >;
C4⋊C4.94D4 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{94}D_4 % in TeX
G:=Group("C4:C4.94D4"); // GroupNames label
G:=SmallGroup(128,774);
// by ID
G=gap.SmallGroup(128,774);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2804,1411,718,172,4037,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations