p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.173D4, C24.334C23, C23.463C24, C22.2482+ 1+4, (C2×D4)⋊17Q8, C42⋊8C4⋊43C2, C23.23(C2×Q8), C23⋊Q8⋊20C2, C4.56(C22⋊Q8), C2.34(D4⋊3Q8), C4.60(C4.4D4), C23.7Q8⋊70C2, (C23×C4).404C22, (C22×C4).100C23, (C2×C42).564C22, C22.314(C22×D4), C22.104(C22×Q8), (C22×D4).532C22, (C22×Q8).139C22, C2.24(C22.29C24), C24.3C22.48C2, C2.C42.199C22, C2.26(C22.49C24), (C4×C4⋊C4)⋊97C2, (C2×C4⋊Q8)⋊15C2, (C2×C4×D4).63C2, (C2×C4).310(C2×Q8), C2.31(C2×C22⋊Q8), (C2×C4).1395(C2×D4), C2.25(C2×C4.4D4), (C2×C4).825(C4○D4), (C2×C4⋊C4).875C22, C22.339(C2×C4○D4), (C2×C22⋊C4).186C22, SmallGroup(128,1295)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.173D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 548 in 282 conjugacy classes, 116 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C4×C4⋊C4, C23.7Q8, C42⋊8C4, C24.3C22, C23⋊Q8, C2×C4×D4, C2×C4⋊Q8, C42.173D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C4.4D4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C2×C4.4D4, C22.29C24, D4⋊3Q8, C22.49C24, C42.173D4
►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 11 20 13)(2 12 17 14)(3 9 18 15)(4 10 19 16)(5 28 63 29)(6 25 64 30)(7 26 61 31)(8 27 62 32)(21 44 52 45)(22 41 49 46)(23 42 50 47)(24 43 51 48)(33 57 39 56)(34 58 40 53)(35 59 37 54)(36 60 38 55)
(1 29 42 53)(2 25 43 59)(3 31 44 55)(4 27 41 57)(5 23 34 13)(6 51 35 12)(7 21 36 15)(8 49 33 10)(9 61 52 38)(11 63 50 40)(14 64 24 37)(16 62 22 39)(17 30 48 54)(18 26 45 60)(19 32 46 56)(20 28 47 58)
(1 33 3 35)(2 36 4 34)(5 43 7 41)(6 42 8 44)(9 54 11 56)(10 53 12 55)(13 57 15 59)(14 60 16 58)(17 38 19 40)(18 37 20 39)(21 25 23 27)(22 28 24 26)(29 51 31 49)(30 50 32 52)(45 64 47 62)(46 63 48 61)
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), (1,11,20,13)(2,12,17,14)(3,9,18,15)(4,10,19,16)(5,28,63,29)(6,25,64,30)(7,26,61,31)(8,27,62,32)(21,44,52,45)(22,41,49,46)(23,42,50,47)(24,43,51,48)(33,57,39,56)(34,58,40,53)(35,59,37,54)(36,60,38,55), (1,29,42,53)(2,25,43,59)(3,31,44,55)(4,27,41,57)(5,23,34,13)(6,51,35,12)(7,21,36,15)(8,49,33,10)(9,61,52,38)(11,63,50,40)(14,64,24,37)(16,62,22,39)(17,30,48,54)(18,26,45,60)(19,32,46,56)(20,28,47,58), (1,33,3,35)(2,36,4,34)(5,43,7,41)(6,42,8,44)(9,54,11,56)(10,53,12,55)(13,57,15,59)(14,60,16,58)(17,38,19,40)(18,37,20,39)(21,25,23,27)(22,28,24,26)(29,51,31,49)(30,50,32,52)(45,64,47,62)(46,63,48,61)>;
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), (1,11,20,13)(2,12,17,14)(3,9,18,15)(4,10,19,16)(5,28,63,29)(6,25,64,30)(7,26,61,31)(8,27,62,32)(21,44,52,45)(22,41,49,46)(23,42,50,47)(24,43,51,48)(33,57,39,56)(34,58,40,53)(35,59,37,54)(36,60,38,55), (1,29,42,53)(2,25,43,59)(3,31,44,55)(4,27,41,57)(5,23,34,13)(6,51,35,12)(7,21,36,15)(8,49,33,10)(9,61,52,38)(11,63,50,40)(14,64,24,37)(16,62,22,39)(17,30,48,54)(18,26,45,60)(19,32,46,56)(20,28,47,58), (1,33,3,35)(2,36,4,34)(5,43,7,41)(6,42,8,44)(9,54,11,56)(10,53,12,55)(13,57,15,59)(14,60,16,58)(17,38,19,40)(18,37,20,39)(21,25,23,27)(22,28,24,26)(29,51,31,49)(30,50,32,52)(45,64,47,62)(46,63,48,61) );
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)], [(1,11,20,13),(2,12,17,14),(3,9,18,15),(4,10,19,16),(5,28,63,29),(6,25,64,30),(7,26,61,31),(8,27,62,32),(21,44,52,45),(22,41,49,46),(23,42,50,47),(24,43,51,48),(33,57,39,56),(34,58,40,53),(35,59,37,54),(36,60,38,55)], [(1,29,42,53),(2,25,43,59),(3,31,44,55),(4,27,41,57),(5,23,34,13),(6,51,35,12),(7,21,36,15),(8,49,33,10),(9,61,52,38),(11,63,50,40),(14,64,24,37),(16,62,22,39),(17,30,48,54),(18,26,45,60),(19,32,46,56),(20,28,47,58)], [(1,33,3,35),(2,36,4,34),(5,43,7,41),(6,42,8,44),(9,54,11,56),(10,53,12,55),(13,57,15,59),(14,60,16,58),(17,38,19,40),(18,37,20,39),(21,25,23,27),(22,28,24,26),(29,51,31,49),(30,50,32,52),(45,64,47,62),(46,63,48,61)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ 1+4 |
| kernel | C42.173D4 | C4×C4⋊C4 | C23.7Q8 | C42⋊8C4 | C24.3C22 | C23⋊Q8 | C2×C4×D4 | C2×C4⋊Q8 | C42 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42.173D4 ►in GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
C42.173D4 in GAP, Magma, Sage, TeX
C_4^2._{173}D_4 % in TeX
G:=Group("C4^2.173D4"); // GroupNames label
G:=SmallGroup(128,1295);
// by ID
G=gap.SmallGroup(128,1295);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,568,758,723,675,80]);
// 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*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations