p-group, metabelian, nilpotent (class 5), monomial
Aliases: C4○D32, C4○Q64, C4○SD64, D32⋊3C2, Q64⋊3C2, C8.22D8, SD64⋊3C2, C4.20D16, C16.13D4, C32.6C22, C16.9C23, C22.1D16, D16.2C22, Q32.2C22, (C2×C32)⋊6C2, C4○D16⋊1C2, C4.16(C2×D8), (C2×C4).91D8, C8.48(C2×D4), C2.15(C2×D16), (C2×C8).271D4, (C2×C16).96C22, SmallGroup(128,994)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4○D32
G = < a,b,c | a4=c2=1, b16=a2, ab=ba, ac=ca, cbc=a2b15 >
Smallest permutation representation of C4○D32
►On 64 points
Generators in S64
(1 48 17 64)(2 49 18 33)(3 50 19 34)(4 51 20 35)(5 52 21 36)(6 53 22 37)(7 54 23 38)(8 55 24 39)(9 56 25 40)(10 57 26 41)(11 58 27 42)(12 59 28 43)(13 60 29 44)(14 61 30 45)(15 62 31 46)(16 63 32 47)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(33 62)(34 61)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(63 64)
G:=sub<Sym(64)| (1,48,17,64)(2,49,18,33)(3,50,19,34)(4,51,20,35)(5,52,21,36)(6,53,22,37)(7,54,23,38)(8,55,24,39)(9,56,25,40)(10,57,26,41)(11,58,27,42)(12,59,28,43)(13,60,29,44)(14,61,30,45)(15,62,31,46)(16,63,32,47), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(63,64)>;
G:=Group( (1,48,17,64)(2,49,18,33)(3,50,19,34)(4,51,20,35)(5,52,21,36)(6,53,22,37)(7,54,23,38)(8,55,24,39)(9,56,25,40)(10,57,26,41)(11,58,27,42)(12,59,28,43)(13,60,29,44)(14,61,30,45)(15,62,31,46)(16,63,32,47), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(63,64) );
G=PermutationGroup([[(1,48,17,64),(2,49,18,33),(3,50,19,34),(4,51,20,35),(5,52,21,36),(6,53,22,37),(7,54,23,38),(8,55,24,39),(9,56,25,40),(10,57,26,41),(11,58,27,42),(12,59,28,43),(13,60,29,44),(14,61,30,45),(15,62,31,46),(16,63,32,47)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(33,62),(34,61),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(63,64)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 16 | 16 | 1 | 1 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | D16 | D16 | C4○D32 |
| kernel | C4○D32 | C2×C32 | D32 | SD64 | Q64 | C4○D16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C4○D32 ►in GL2(𝔽97) generated by
| 22 | 0 |
| 0 | 22 |
| 57 | 70 |
| 27 | 57 |
| 57 | 70 |
| 70 | 40 |
G:=sub<GL(2,GF(97))| [22,0,0,22],[57,27,70,57],[57,70,70,40] >;
C4○D32 in GAP, Magma, Sage, TeX
C_4\circ D_{32} % in TeX
G:=Group("C4oD32"); // GroupNames label
G:=SmallGroup(128,994);
// by ID
G=gap.SmallGroup(128,994);
# by ID
G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,141,352,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^16=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^15>;
// generators/relations
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