p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.97D4, (C2×Q8).15Q8, (C2×C42).21C4, (C2×Q8).156D4, C4⋊2(C4.10D4), C4.92(C22⋊Q8), C4.117(C4⋊D4), C4⋊M4(2).26C2, (C22×C4).661C23, C23.187(C22×C4), (C2×C42).246C22, C22.C42.10C2, C2.10(C23.7Q8), (C22×Q8).382C22, C22.27(C42⋊C2), (C2×M4(2)).152C22, C2.24(M4(2).8C22), (C2×C4⋊C4).51C4, (C2×C4×Q8).10C2, (C2×C4).2(C2×Q8), (C2×C4).10(C4⋊C4), C22.19(C2×C4⋊C4), (C2×C4).46(C4○D4), (C2×C4).1313(C2×D4), (C22×C4).51(C2×C4), (C2×C4⋊C4).754C22, (C2×C4.10D4).6C2, C2.23(C2×C4.10D4), (C2×C4).357(C22⋊C4), C22.246(C2×C22⋊C4), SmallGroup(128,533)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.97D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=bc3 >
Subgroups: 228 in 136 conjugacy classes, 62 normal (20 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×9], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×4], C2×C4 [×10], C2×C4 [×11], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4.10D4 [×4], C4⋊C8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×4], C2×M4(2) [×4], C22×Q8, C22.C42 [×2], C2×C4.10D4 [×2], C4⋊M4(2) [×2], C2×C4×Q8, C42.97D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C4.10D4, M4(2).8C22, C42.97D4
►On 64 points
(1 46 19 37)(2 34 20 43)(3 48 21 39)(4 36 22 45)(5 42 23 33)(6 38 24 47)(7 44 17 35)(8 40 18 41)(9 56 60 31)(10 28 61 53)(11 50 62 25)(12 30 63 55)(13 52 64 27)(14 32 57 49)(15 54 58 29)(16 26 59 51)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 55 53 51)(50 52 54 56)(57 63 61 59)(58 60 62 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 31 7 25 5 27 3 29)(2 32 4 30 6 28 8 26)(9 39 11 37 13 35 15 33)(10 36 16 38 14 40 12 34)(17 50 23 52 21 54 19 56)(18 51 20 49 22 55 24 53)(41 63 43 61 45 59 47 57)(42 60 48 62 46 64 44 58)
G:=sub<Sym(64)| (1,46,19,37)(2,34,20,43)(3,48,21,39)(4,36,22,45)(5,42,23,33)(6,38,24,47)(7,44,17,35)(8,40,18,41)(9,56,60,31)(10,28,61,53)(11,50,62,25)(12,30,63,55)(13,52,64,27)(14,32,57,49)(15,54,58,29)(16,26,59,51), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,55,53,51)(50,52,54,56)(57,63,61,59)(58,60,62,64), (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), (1,31,7,25,5,27,3,29)(2,32,4,30,6,28,8,26)(9,39,11,37,13,35,15,33)(10,36,16,38,14,40,12,34)(17,50,23,52,21,54,19,56)(18,51,20,49,22,55,24,53)(41,63,43,61,45,59,47,57)(42,60,48,62,46,64,44,58)>;
G:=Group( (1,46,19,37)(2,34,20,43)(3,48,21,39)(4,36,22,45)(5,42,23,33)(6,38,24,47)(7,44,17,35)(8,40,18,41)(9,56,60,31)(10,28,61,53)(11,50,62,25)(12,30,63,55)(13,52,64,27)(14,32,57,49)(15,54,58,29)(16,26,59,51), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,55,53,51)(50,52,54,56)(57,63,61,59)(58,60,62,64), (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), (1,31,7,25,5,27,3,29)(2,32,4,30,6,28,8,26)(9,39,11,37,13,35,15,33)(10,36,16,38,14,40,12,34)(17,50,23,52,21,54,19,56)(18,51,20,49,22,55,24,53)(41,63,43,61,45,59,47,57)(42,60,48,62,46,64,44,58) );
G=PermutationGroup([(1,46,19,37),(2,34,20,43),(3,48,21,39),(4,36,22,45),(5,42,23,33),(6,38,24,47),(7,44,17,35),(8,40,18,41),(9,56,60,31),(10,28,61,53),(11,50,62,25),(12,30,63,55),(13,52,64,27),(14,32,57,49),(15,54,58,29),(16,26,59,51)], [(1,7,5,3),(2,4,6,8),(9,11,13,15),(10,16,14,12),(17,23,21,19),(18,20,22,24),(25,27,29,31),(26,32,30,28),(33,39,37,35),(34,36,38,40),(41,43,45,47),(42,48,46,44),(49,55,53,51),(50,52,54,56),(57,63,61,59),(58,60,62,64)], [(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)], [(1,31,7,25,5,27,3,29),(2,32,4,30,6,28,8,26),(9,39,11,37,13,35,15,33),(10,36,16,38,14,40,12,34),(17,50,23,52,21,54,19,56),(18,51,20,49,22,55,24,53),(41,63,43,61,45,59,47,57),(42,60,48,62,46,64,44,58)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | Q8 | C4○D4 | C4.10D4 | M4(2).8C22 |
| kernel | C42.97D4 | C22.C42 | C2×C4.10D4 | C4⋊M4(2) | C2×C4×Q8 | C2×C42 | C2×C4⋊C4 | C42 | C2×Q8 | C2×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C42.97D4 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 1 |
| 0 | 0 | 0 | 11 | 16 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 3 | 2 |
| 0 | 0 | 1 | 8 | 2 | 14 |
| 0 | 0 | 2 | 5 | 13 | 16 |
| 0 | 0 | 5 | 14 | 15 | 9 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 11 | 15 | 0 |
| 0 | 0 | 11 | 7 | 0 | 15 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 1 | 0 | 6 | 10 |
G:=sub<GL(6,GF(17))| [13,9,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,6,0,0,0,1,0,0,11,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,13,13,0,0,0,0,0,0,4,1,2,5,0,0,2,8,5,14,0,0,3,2,13,15,0,0,2,14,16,9],[13,9,0,0,0,0,4,4,0,0,0,0,0,0,7,11,0,1,0,0,11,7,0,0,0,0,15,0,10,6,0,0,0,15,6,10] >;
C42.97D4 in GAP, Magma, Sage, TeX
C_4^2._{97}D_4 % in TeX
G:=Group("C4^2.97D4"); // GroupNames label
G:=SmallGroup(128,533);
// by ID
G=gap.SmallGroup(128,533);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,352,2804,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations