p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42.196D4, C24.53C23, C23.558C24, C22.2482- (1+4), C4.8(C4⋊1D4), C42⋊4C4⋊34C2, C23.4Q8⋊36C2, (C2×C42).623C22, (C22×C4).857C23, C22.370(C22×D4), (C22×D4).208C22, (C22×Q8).165C22, C2.C42.561C22, C2.47(C23.38C23), (C2×C4⋊Q8)⋊19C2, (C2×C4).404(C2×D4), C2.15(C2×C4⋊1D4), (C2×C4⋊C4).381C22, (C2×C4.4D4).29C2, (C2×C22⋊C4).238C22, SmallGroup(128,1390)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 580 in 306 conjugacy classes, 116 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×18], C22 [×7], C22 [×14], C2×C4 [×18], C2×C4 [×30], D4 [×4], Q8 [×12], C23, C23 [×14], C42 [×12], C22⋊C4 [×24], C4⋊C4 [×24], C22×C4, C22×C4 [×12], C2×D4 [×6], C2×Q8 [×18], C24 [×2], C2.C42 [×4], C2×C42 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×12], C4.4D4 [×12], C4⋊Q8 [×12], C22×D4, C22×Q8 [×3], C42⋊4C4, C23.4Q8 [×8], C2×C4.4D4 [×3], C2×C4⋊Q8 [×3], C42.196D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C22×D4 [×3], 2- (1+4) [×4], C2×C4⋊1D4, C23.38C23 [×6], C42.196D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
(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 24 54)(2 8 21 55)(3 5 22 56)(4 6 23 53)(9 44 47 37)(10 41 48 38)(11 42 45 39)(12 43 46 40)(13 26 59 30)(14 27 60 31)(15 28 57 32)(16 25 58 29)(17 52 34 62)(18 49 35 63)(19 50 36 64)(20 51 33 61)
(1 28 51 12)(2 29 52 47)(3 26 49 10)(4 31 50 45)(5 59 35 41)(6 14 36 39)(7 57 33 43)(8 16 34 37)(9 21 25 62)(11 23 27 64)(13 18 38 56)(15 20 40 54)(17 44 55 58)(19 42 53 60)(22 30 63 48)(24 32 61 46)
(1 46 24 12)(2 45 21 11)(3 48 22 10)(4 47 23 9)(5 41 56 38)(6 44 53 37)(7 43 54 40)(8 42 55 39)(13 35 59 18)(14 34 60 17)(15 33 57 20)(16 36 58 19)(25 50 29 64)(26 49 30 63)(27 52 31 62)(28 51 32 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), (1,7,24,54)(2,8,21,55)(3,5,22,56)(4,6,23,53)(9,44,47,37)(10,41,48,38)(11,42,45,39)(12,43,46,40)(13,26,59,30)(14,27,60,31)(15,28,57,32)(16,25,58,29)(17,52,34,62)(18,49,35,63)(19,50,36,64)(20,51,33,61), (1,28,51,12)(2,29,52,47)(3,26,49,10)(4,31,50,45)(5,59,35,41)(6,14,36,39)(7,57,33,43)(8,16,34,37)(9,21,25,62)(11,23,27,64)(13,18,38,56)(15,20,40,54)(17,44,55,58)(19,42,53,60)(22,30,63,48)(24,32,61,46), (1,46,24,12)(2,45,21,11)(3,48,22,10)(4,47,23,9)(5,41,56,38)(6,44,53,37)(7,43,54,40)(8,42,55,39)(13,35,59,18)(14,34,60,17)(15,33,57,20)(16,36,58,19)(25,50,29,64)(26,49,30,63)(27,52,31,62)(28,51,32,61)>;
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), (1,7,24,54)(2,8,21,55)(3,5,22,56)(4,6,23,53)(9,44,47,37)(10,41,48,38)(11,42,45,39)(12,43,46,40)(13,26,59,30)(14,27,60,31)(15,28,57,32)(16,25,58,29)(17,52,34,62)(18,49,35,63)(19,50,36,64)(20,51,33,61), (1,28,51,12)(2,29,52,47)(3,26,49,10)(4,31,50,45)(5,59,35,41)(6,14,36,39)(7,57,33,43)(8,16,34,37)(9,21,25,62)(11,23,27,64)(13,18,38,56)(15,20,40,54)(17,44,55,58)(19,42,53,60)(22,30,63,48)(24,32,61,46), (1,46,24,12)(2,45,21,11)(3,48,22,10)(4,47,23,9)(5,41,56,38)(6,44,53,37)(7,43,54,40)(8,42,55,39)(13,35,59,18)(14,34,60,17)(15,33,57,20)(16,36,58,19)(25,50,29,64)(26,49,30,63)(27,52,31,62)(28,51,32,61) );
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)], [(1,7,24,54),(2,8,21,55),(3,5,22,56),(4,6,23,53),(9,44,47,37),(10,41,48,38),(11,42,45,39),(12,43,46,40),(13,26,59,30),(14,27,60,31),(15,28,57,32),(16,25,58,29),(17,52,34,62),(18,49,35,63),(19,50,36,64),(20,51,33,61)], [(1,28,51,12),(2,29,52,47),(3,26,49,10),(4,31,50,45),(5,59,35,41),(6,14,36,39),(7,57,33,43),(8,16,34,37),(9,21,25,62),(11,23,27,64),(13,18,38,56),(15,20,40,54),(17,44,55,58),(19,42,53,60),(22,30,63,48),(24,32,61,46)], [(1,46,24,12),(2,45,21,11),(3,48,22,10),(4,47,23,9),(5,41,56,38),(6,44,53,37),(7,43,54,40),(8,42,55,39),(13,35,59,18),(14,34,60,17),(15,33,57,20),(16,36,58,19),(25,50,29,64),(26,49,30,63),(27,52,31,62),(28,51,32,61)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 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 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 4 | 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 | 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 | 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 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0] >;
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 | 2 | 4 |
| type | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | D4 | 2- (1+4) |
| kernel | C42.196D4 | C42⋊4C4 | C23.4Q8 | C2×C4.4D4 | C2×C4⋊Q8 | C42 | C22 |
| # reps | 1 | 1 | 8 | 3 | 3 | 12 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{196}D_4 % in TeX
G:=Group("C4^2.196D4"); // GroupNames label
G:=SmallGroup(128,1390);
// by ID
G=gap.SmallGroup(128,1390);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,568,758,723,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^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=b^2*c^-1>;
// generators/relations