p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.193D4, C24.42C23, C23.540C24, C22.3162+ 1+4, C22.2332- 1+4, C42⋊4C4⋊30C2, C23⋊Q8⋊33C2, C4.13(C4.4D4), C23.11D4⋊64C2, (C22×C4).150C23, (C2×C42).616C22, C22.365(C22×D4), (C22×D4).198C22, (C22×Q8).159C22, C23.67C23⋊73C2, C2.45(C22.29C24), C24.3C22.56C2, C2.C42.553C22, C2.52(C22.36C24), C2.45(C23.38C23), (C2×C4⋊Q8)⋊18C2, (C2×C4).399(C2×D4), C2.30(C2×C4.4D4), (C2×C4).662(C4○D4), (C2×C4⋊C4).366C22, (C2×C4.4D4).28C2, C22.412(C2×C4○D4), (C2×C22⋊C4).228C22, SmallGroup(128,1372)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.193D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c-1 >
Subgroups: 516 in 250 conjugacy classes, 100 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C4⋊Q8, C22×D4, C22×Q8, C22×Q8, C42⋊4C4, C24.3C22, C23.67C23, C23⋊Q8, C23.11D4, C2×C4.4D4, C2×C4⋊Q8, C42.193D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4.4D4, C22.29C24, C23.38C23, C22.36C24, C42.193D4
►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 7 19 54)(2 8 20 55)(3 5 17 56)(4 6 18 53)(9 43 47 31)(10 44 48 32)(11 41 45 29)(12 42 46 30)(13 40 59 26)(14 37 60 27)(15 38 57 28)(16 39 58 25)(21 51 33 61)(22 52 34 62)(23 49 35 63)(24 50 36 64)
(1 40 49 12)(2 27 50 47)(3 38 51 10)(4 25 52 45)(5 57 33 44)(6 16 34 29)(7 59 35 42)(8 14 36 31)(9 20 37 64)(11 18 39 62)(13 23 30 54)(15 21 32 56)(17 28 61 48)(19 26 63 46)(22 41 53 58)(24 43 55 60)
(1 57 17 13)(2 60 18 16)(3 59 19 15)(4 58 20 14)(5 40 54 28)(6 39 55 27)(7 38 56 26)(8 37 53 25)(9 22 45 36)(10 21 46 35)(11 24 47 34)(12 23 48 33)(29 50 43 62)(30 49 44 61)(31 52 41 64)(32 51 42 63)
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), (1,7,19,54)(2,8,20,55)(3,5,17,56)(4,6,18,53)(9,43,47,31)(10,44,48,32)(11,41,45,29)(12,42,46,30)(13,40,59,26)(14,37,60,27)(15,38,57,28)(16,39,58,25)(21,51,33,61)(22,52,34,62)(23,49,35,63)(24,50,36,64), (1,40,49,12)(2,27,50,47)(3,38,51,10)(4,25,52,45)(5,57,33,44)(6,16,34,29)(7,59,35,42)(8,14,36,31)(9,20,37,64)(11,18,39,62)(13,23,30,54)(15,21,32,56)(17,28,61,48)(19,26,63,46)(22,41,53,58)(24,43,55,60), (1,57,17,13)(2,60,18,16)(3,59,19,15)(4,58,20,14)(5,40,54,28)(6,39,55,27)(7,38,56,26)(8,37,53,25)(9,22,45,36)(10,21,46,35)(11,24,47,34)(12,23,48,33)(29,50,43,62)(30,49,44,61)(31,52,41,64)(32,51,42,63)>;
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), (1,7,19,54)(2,8,20,55)(3,5,17,56)(4,6,18,53)(9,43,47,31)(10,44,48,32)(11,41,45,29)(12,42,46,30)(13,40,59,26)(14,37,60,27)(15,38,57,28)(16,39,58,25)(21,51,33,61)(22,52,34,62)(23,49,35,63)(24,50,36,64), (1,40,49,12)(2,27,50,47)(3,38,51,10)(4,25,52,45)(5,57,33,44)(6,16,34,29)(7,59,35,42)(8,14,36,31)(9,20,37,64)(11,18,39,62)(13,23,30,54)(15,21,32,56)(17,28,61,48)(19,26,63,46)(22,41,53,58)(24,43,55,60), (1,57,17,13)(2,60,18,16)(3,59,19,15)(4,58,20,14)(5,40,54,28)(6,39,55,27)(7,38,56,26)(8,37,53,25)(9,22,45,36)(10,21,46,35)(11,24,47,34)(12,23,48,33)(29,50,43,62)(30,49,44,61)(31,52,41,64)(32,51,42,63) );
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)], [(1,7,19,54),(2,8,20,55),(3,5,17,56),(4,6,18,53),(9,43,47,31),(10,44,48,32),(11,41,45,29),(12,42,46,30),(13,40,59,26),(14,37,60,27),(15,38,57,28),(16,39,58,25),(21,51,33,61),(22,52,34,62),(23,49,35,63),(24,50,36,64)], [(1,40,49,12),(2,27,50,47),(3,38,51,10),(4,25,52,45),(5,57,33,44),(6,16,34,29),(7,59,35,42),(8,14,36,31),(9,20,37,64),(11,18,39,62),(13,23,30,54),(15,21,32,56),(17,28,61,48),(19,26,63,46),(22,41,53,58),(24,43,55,60)], [(1,57,17,13),(2,60,18,16),(3,59,19,15),(4,58,20,14),(5,40,54,28),(6,39,55,27),(7,38,56,26),(8,37,53,25),(9,22,45,36),(10,21,46,35),(11,24,47,34),(12,23,48,33),(29,50,43,62),(30,49,44,61),(31,52,41,64),(32,51,42,63)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.193D4 | C42⋊4C4 | C24.3C22 | C23.67C23 | C23⋊Q8 | C23.11D4 | C2×C4.4D4 | C2×C4⋊Q8 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of C42.193D4 ►in GL8(𝔽5)
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 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 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 |
| 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 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 | 0 | 0 |
G:=sub<GL(8,GF(5))| [0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,3,0,0,0,0,3,3,0,0,0,0,0,0,4,2,0,0],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,3,0,0,0,0,2,2,0,0,0,0,0,0,0,3,0,0] >;
C42.193D4 in GAP, Magma, Sage, TeX
C_4^2._{193}D_4 % in TeX
G:=Group("C4^2.193D4"); // GroupNames label
G:=SmallGroup(128,1372);
// by ID
G=gap.SmallGroup(128,1372);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,185,80]);
// 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*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations