p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.198D4, C24.55C23, C23.564C24, C22.3382+ 1+4, C22.2532- 1+4, (C2×C42).628C22, (C22×C4).169C23, C22.376(C22×D4), C23.4Q8.17C2, C23.Q8.24C2, C23.11D4.32C2, C23.83C23⋊72C2, C23.81C23⋊73C2, C24.C22.46C2, C23.65C23⋊111C2, C2.C42.278C22, C2.53(C22.26C24), C2.50(C23.38C23), C2.54(C22.33C24), C2.35(C22.31C24), C2.36(C22.35C24), (C4×C4⋊C4)⋊116C2, (C2×C4).408(C2×D4), (C2×C42.C2)⋊19C2, (C2×C4).184(C4○D4), (C2×C4⋊C4).897C22, C22.431(C2×C4○D4), (C2×C42⋊2C2).11C2, (C2×C22⋊C4).241C22, SmallGroup(128,1396)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.198D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 388 in 217 conjugacy classes, 96 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42⋊2C2, C4×C4⋊C4, C24.C22, C23.65C23, C23.Q8, C23.11D4, C23.81C23, C23.81C23, C23.4Q8, C23.83C23, C2×C42.C2, C2×C42⋊2C2, C42.198D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.26C24, C23.38C23, C22.31C24, C22.33C24, C22.35C24, C42.198D4
►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 26 59 36)(2 27 60 33)(3 28 57 34)(4 25 58 35)(5 64 32 56)(6 61 29 53)(7 62 30 54)(8 63 31 55)(9 37 46 17)(10 38 47 18)(11 39 48 19)(12 40 45 20)(13 42 50 24)(14 43 51 21)(15 44 52 22)(16 41 49 23)
(1 48 29 22)(2 47 30 21)(3 46 31 24)(4 45 32 23)(5 41 58 12)(6 44 59 11)(7 43 60 10)(8 42 57 9)(13 28 17 55)(14 27 18 54)(15 26 19 53)(16 25 20 56)(33 38 62 51)(34 37 63 50)(35 40 64 49)(36 39 61 52)
(1 27 3 25)(2 36 4 34)(5 55 7 53)(6 62 8 64)(9 49 11 51)(10 15 12 13)(14 46 16 48)(17 43 19 41)(18 24 20 22)(21 39 23 37)(26 58 28 60)(29 54 31 56)(30 61 32 63)(33 57 35 59)(38 42 40 44)(45 50 47 52)
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), (1,26,59,36)(2,27,60,33)(3,28,57,34)(4,25,58,35)(5,64,32,56)(6,61,29,53)(7,62,30,54)(8,63,31,55)(9,37,46,17)(10,38,47,18)(11,39,48,19)(12,40,45,20)(13,42,50,24)(14,43,51,21)(15,44,52,22)(16,41,49,23), (1,48,29,22)(2,47,30,21)(3,46,31,24)(4,45,32,23)(5,41,58,12)(6,44,59,11)(7,43,60,10)(8,42,57,9)(13,28,17,55)(14,27,18,54)(15,26,19,53)(16,25,20,56)(33,38,62,51)(34,37,63,50)(35,40,64,49)(36,39,61,52), (1,27,3,25)(2,36,4,34)(5,55,7,53)(6,62,8,64)(9,49,11,51)(10,15,12,13)(14,46,16,48)(17,43,19,41)(18,24,20,22)(21,39,23,37)(26,58,28,60)(29,54,31,56)(30,61,32,63)(33,57,35,59)(38,42,40,44)(45,50,47,52)>;
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), (1,26,59,36)(2,27,60,33)(3,28,57,34)(4,25,58,35)(5,64,32,56)(6,61,29,53)(7,62,30,54)(8,63,31,55)(9,37,46,17)(10,38,47,18)(11,39,48,19)(12,40,45,20)(13,42,50,24)(14,43,51,21)(15,44,52,22)(16,41,49,23), (1,48,29,22)(2,47,30,21)(3,46,31,24)(4,45,32,23)(5,41,58,12)(6,44,59,11)(7,43,60,10)(8,42,57,9)(13,28,17,55)(14,27,18,54)(15,26,19,53)(16,25,20,56)(33,38,62,51)(34,37,63,50)(35,40,64,49)(36,39,61,52), (1,27,3,25)(2,36,4,34)(5,55,7,53)(6,62,8,64)(9,49,11,51)(10,15,12,13)(14,46,16,48)(17,43,19,41)(18,24,20,22)(21,39,23,37)(26,58,28,60)(29,54,31,56)(30,61,32,63)(33,57,35,59)(38,42,40,44)(45,50,47,52) );
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)], [(1,26,59,36),(2,27,60,33),(3,28,57,34),(4,25,58,35),(5,64,32,56),(6,61,29,53),(7,62,30,54),(8,63,31,55),(9,37,46,17),(10,38,47,18),(11,39,48,19),(12,40,45,20),(13,42,50,24),(14,43,51,21),(15,44,52,22),(16,41,49,23)], [(1,48,29,22),(2,47,30,21),(3,46,31,24),(4,45,32,23),(5,41,58,12),(6,44,59,11),(7,43,60,10),(8,42,57,9),(13,28,17,55),(14,27,18,54),(15,26,19,53),(16,25,20,56),(33,38,62,51),(34,37,63,50),(35,40,64,49),(36,39,61,52)], [(1,27,3,25),(2,36,4,34),(5,55,7,53),(6,62,8,64),(9,49,11,51),(10,15,12,13),(14,46,16,48),(17,43,19,41),(18,24,20,22),(21,39,23,37),(26,58,28,60),(29,54,31,56),(30,61,32,63),(33,57,35,59),(38,42,40,44),(45,50,47,52)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4W |
| order | 1 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.198D4 | C4×C4⋊C4 | C24.C22 | C23.65C23 | C23.Q8 | C23.11D4 | C23.81C23 | C23.4Q8 | C23.83C23 | C2×C42.C2 | C2×C42⋊2C2 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 3 | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 3 |
Matrix representation of C42.198D4 ►in GL8(𝔽5)
| 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 2 | 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 | 0 | 4 | 1 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 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 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 2 | 3 |
| 0 | 0 | 0 | 0 | 2 | 1 | 4 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 4 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 1 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 1 |
G:=sub<GL(8,GF(5))| [3,2,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,2,2,0,0,0,0,0,4,0,3,3],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,2,0,0,0,0,4,0,1,1,0,0,0,0,0,0,2,4,0,0,0,0,0,0,3,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,4,2,3,1,0,0,0,0,0,3,0,2],[4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,1,1,0,0,0,0,0,4,0,2,0,0,0,0,3,1,1,0,0,0,0,0,0,4,0,1] >;
C42.198D4 in GAP, Magma, Sage, TeX
C_4^2._{198}D_4 % in TeX
G:=Group("C4^2.198D4"); // GroupNames label
G:=SmallGroup(128,1396);
// by ID
G=gap.SmallGroup(128,1396);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations