p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.194D4, C24.43C23, C23.541C24, C22.3172+ 1+4, C42⋊4C4⋊31C2, C4.14(C4.4D4), C23.10D4⋊64C2, (C22×C4).151C23, (C2×C42).617C22, C22.366(C22×D4), C24.3C22⋊67C2, (C22×D4).199C22, C2.46(C22.29C24), C2.C42.554C22, C2.29(C22.34C24), (C2×C4).400(C2×D4), (C2×C4⋊1D4).18C2, (C2×C42.C2)⋊18C2, C2.31(C2×C4.4D4), (C2×C4).663(C4○D4), (C2×C4⋊C4).367C22, C22.413(C2×C4○D4), (C2×C22⋊C4).229C22, SmallGroup(128,1373)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.194D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2b2c-1 >
Subgroups: 676 in 292 conjugacy classes, 100 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42.C2, C4⋊1D4, C22×D4, C42⋊4C4, C24.3C22, C23.10D4, C2×C42.C2, C2×C4⋊1D4, C42.194D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4.4D4, C22.29C24, C22.34C24, C42.194D4
►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 50 19 63)(2 51 20 64)(3 52 17 61)(4 49 18 62)(5 38 46 26)(6 39 47 27)(7 40 48 28)(8 37 45 25)(9 29 54 44)(10 30 55 41)(11 31 56 42)(12 32 53 43)(13 33 58 22)(14 34 59 23)(15 35 60 24)(16 36 57 21)
(1 29 14 46)(2 41 15 6)(3 31 16 48)(4 43 13 8)(5 19 44 59)(7 17 42 57)(9 23 38 63)(10 35 39 51)(11 21 40 61)(12 33 37 49)(18 32 58 45)(20 30 60 47)(22 25 62 53)(24 27 64 55)(26 50 54 34)(28 52 56 36)
(1 11 17 54)(2 55 18 12)(3 9 19 56)(4 53 20 10)(5 23 48 36)(6 33 45 24)(7 21 46 34)(8 35 47 22)(13 25 60 39)(14 40 57 26)(15 27 58 37)(16 38 59 28)(29 50 42 61)(30 62 43 51)(31 52 44 63)(32 64 41 49)
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), (1,50,19,63)(2,51,20,64)(3,52,17,61)(4,49,18,62)(5,38,46,26)(6,39,47,27)(7,40,48,28)(8,37,45,25)(9,29,54,44)(10,30,55,41)(11,31,56,42)(12,32,53,43)(13,33,58,22)(14,34,59,23)(15,35,60,24)(16,36,57,21), (1,29,14,46)(2,41,15,6)(3,31,16,48)(4,43,13,8)(5,19,44,59)(7,17,42,57)(9,23,38,63)(10,35,39,51)(11,21,40,61)(12,33,37,49)(18,32,58,45)(20,30,60,47)(22,25,62,53)(24,27,64,55)(26,50,54,34)(28,52,56,36), (1,11,17,54)(2,55,18,12)(3,9,19,56)(4,53,20,10)(5,23,48,36)(6,33,45,24)(7,21,46,34)(8,35,47,22)(13,25,60,39)(14,40,57,26)(15,27,58,37)(16,38,59,28)(29,50,42,61)(30,62,43,51)(31,52,44,63)(32,64,41,49)>;
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), (1,50,19,63)(2,51,20,64)(3,52,17,61)(4,49,18,62)(5,38,46,26)(6,39,47,27)(7,40,48,28)(8,37,45,25)(9,29,54,44)(10,30,55,41)(11,31,56,42)(12,32,53,43)(13,33,58,22)(14,34,59,23)(15,35,60,24)(16,36,57,21), (1,29,14,46)(2,41,15,6)(3,31,16,48)(4,43,13,8)(5,19,44,59)(7,17,42,57)(9,23,38,63)(10,35,39,51)(11,21,40,61)(12,33,37,49)(18,32,58,45)(20,30,60,47)(22,25,62,53)(24,27,64,55)(26,50,54,34)(28,52,56,36), (1,11,17,54)(2,55,18,12)(3,9,19,56)(4,53,20,10)(5,23,48,36)(6,33,45,24)(7,21,46,34)(8,35,47,22)(13,25,60,39)(14,40,57,26)(15,27,58,37)(16,38,59,28)(29,50,42,61)(30,62,43,51)(31,52,44,63)(32,64,41,49) );
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)], [(1,50,19,63),(2,51,20,64),(3,52,17,61),(4,49,18,62),(5,38,46,26),(6,39,47,27),(7,40,48,28),(8,37,45,25),(9,29,54,44),(10,30,55,41),(11,31,56,42),(12,32,53,43),(13,33,58,22),(14,34,59,23),(15,35,60,24),(16,36,57,21)], [(1,29,14,46),(2,41,15,6),(3,31,16,48),(4,43,13,8),(5,19,44,59),(7,17,42,57),(9,23,38,63),(10,35,39,51),(11,21,40,61),(12,33,37,49),(18,32,58,45),(20,30,60,47),(22,25,62,53),(24,27,64,55),(26,50,54,34),(28,52,56,36)], [(1,11,17,54),(2,55,18,12),(3,9,19,56),(4,53,20,10),(5,23,48,36),(6,33,45,24),(7,21,46,34),(8,35,47,22),(13,25,60,39),(14,40,57,26),(15,27,58,37),(16,38,59,28),(29,50,42,61),(30,62,43,51),(31,52,44,63),(32,64,41,49)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R | 4S | 4T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 |
| kernel | C42.194D4 | C42⋊4C4 | C24.3C22 | C23.10D4 | C2×C42.C2 | C2×C4⋊1D4 | C42 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 4 | 8 | 4 |
Matrix representation of C42.194D4 ►in GL8(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 4 | 1 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 |
| 0 | 0 | 0 | 0 | 3 | 2 | 4 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 3 | 4 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,3,3,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,3,4,2,2],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,3,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,2,1,3,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,1,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1] >;
C42.194D4 in GAP, Magma, Sage, TeX
C_4^2._{194}D_4 % in TeX
G:=Group("C4^2.194D4"); // GroupNames label
G:=SmallGroup(128,1373);
// by ID
G=gap.SmallGroup(128,1373);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations