p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43⋊9C2, C42⋊36D4, C23.175C24, C24.185C23, (C4×D4)⋊18C4, C4.181(C4×D4), C4⋊1(C42⋊C2), C42.277(C2×C4), C23.75(C22×C4), C22.66(C23×C4), C22.73(C22×D4), C4○(C24.3C22), (C23×C4).281C22, C4○2(C24.C22), C4○(C23.65C23), (C2×C42).1004C22, (C22×C4).1238C23, (C22×D4).465C22, C24.C22⋊189C2, C24.3C22.83C2, C2.2(C22.26C24), C23.65C23⋊172C2, C2.C42.510C22, C2.3(C23.36C23), (C4×C4⋊C4)⋊16C2, C2.12(C2×C4×D4), (C2×C4×D4).26C2, C2.6(C4×C4○D4), C4⋊C4.198(C2×C4), (C4×C22⋊C4)⋊25C2, (C2×C4).674(C2×D4), (C2×C42⋊C2)⋊6C2, (C2×D4).209(C2×C4), C22⋊C4.52(C2×C4), C22.67(C2×C4○D4), (C2×C4).635(C4○D4), (C2×C4⋊C4).790C22, (C2×C4).208(C22×C4), (C22×C4).294(C2×C4), C2.16(C2×C42⋊C2), (C2×C22⋊C4).416C22, SmallGroup(128,1025)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43⋊9C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1, cd=dc >
Subgroups: 524 in 326 conjugacy classes, 160 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C23×C4, C22×D4, C43, C4×C22⋊C4, C4×C4⋊C4, C24.C22, C23.65C23, C24.3C22, C2×C42⋊C2, C2×C4×D4, C43⋊9C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C42⋊C2, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C42⋊C2, C2×C4×D4, C4×C4○D4, C23.36C23, C22.26C24, C43⋊9C2
►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 59 51 45)(2 60 52 46)(3 57 49 47)(4 58 50 48)(5 44 62 54)(6 41 63 55)(7 42 64 56)(8 43 61 53)(9 31 23 17)(10 32 24 18)(11 29 21 19)(12 30 22 20)(13 33 27 39)(14 34 28 40)(15 35 25 37)(16 36 26 38)
(1 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(1 45)(2 18)(3 47)(4 20)(5 16)(6 41)(7 14)(8 43)(9 17)(10 46)(11 19)(12 48)(13 39)(15 37)(21 29)(22 58)(23 31)(24 60)(25 35)(26 62)(27 33)(28 64)(30 50)(32 52)(34 56)(36 54)(38 44)(40 42)(49 57)(51 59)(53 61)(55 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), (1,59,51,45)(2,60,52,46)(3,57,49,47)(4,58,50,48)(5,44,62,54)(6,41,63,55)(7,42,64,56)(8,43,61,53)(9,31,23,17)(10,32,24,18)(11,29,21,19)(12,30,22,20)(13,33,27,39)(14,34,28,40)(15,35,25,37)(16,36,26,38), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,45)(2,18)(3,47)(4,20)(5,16)(6,41)(7,14)(8,43)(9,17)(10,46)(11,19)(12,48)(13,39)(15,37)(21,29)(22,58)(23,31)(24,60)(25,35)(26,62)(27,33)(28,64)(30,50)(32,52)(34,56)(36,54)(38,44)(40,42)(49,57)(51,59)(53,61)(55,63)>;
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), (1,59,51,45)(2,60,52,46)(3,57,49,47)(4,58,50,48)(5,44,62,54)(6,41,63,55)(7,42,64,56)(8,43,61,53)(9,31,23,17)(10,32,24,18)(11,29,21,19)(12,30,22,20)(13,33,27,39)(14,34,28,40)(15,35,25,37)(16,36,26,38), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,45)(2,18)(3,47)(4,20)(5,16)(6,41)(7,14)(8,43)(9,17)(10,46)(11,19)(12,48)(13,39)(15,37)(21,29)(22,58)(23,31)(24,60)(25,35)(26,62)(27,33)(28,64)(30,50)(32,52)(34,56)(36,54)(38,44)(40,42)(49,57)(51,59)(53,61)(55,63) );
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)], [(1,59,51,45),(2,60,52,46),(3,57,49,47),(4,58,50,48),(5,44,62,54),(6,41,63,55),(7,42,64,56),(8,43,61,53),(9,31,23,17),(10,32,24,18),(11,29,21,19),(12,30,22,20),(13,33,27,39),(14,34,28,40),(15,35,25,37),(16,36,26,38)], [(1,13,9,41),(2,14,10,42),(3,15,11,43),(4,16,12,44),(5,48,38,20),(6,45,39,17),(7,46,40,18),(8,47,37,19),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,35),(30,62,58,36),(31,63,59,33),(32,64,60,34)], [(1,45),(2,18),(3,47),(4,20),(5,16),(6,41),(7,14),(8,43),(9,17),(10,46),(11,19),(12,48),(13,39),(15,37),(21,29),(22,58),(23,31),(24,60),(25,35),(26,62),(27,33),(28,64),(30,50),(32,52),(34,56),(36,54),(38,44),(40,42),(49,57),(51,59),(53,61),(55,63)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AF | 4AG | ··· | 4AR |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 |
| kernel | C43⋊9C2 | C43 | C4×C22⋊C4 | C4×C4⋊C4 | C24.C22 | C23.65C23 | C24.3C22 | C2×C42⋊C2 | C2×C4×D4 | C4×D4 | C42 | C2×C4 |
| # reps | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 16 | 4 | 20 |
Matrix representation of C43⋊9C2 ►in GL5(𝔽5)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,4] >;
C43⋊9C2 in GAP, Magma, Sage, TeX
C_4^3\rtimes_9C_2
% in TeX
G:=Group("C4^3:9C2"); // GroupNames label
G:=SmallGroup(128,1025);
// by ID
G=gap.SmallGroup(128,1025);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,184,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*c^2,b*c=c*b,d*b*d=b^-1,c*d=d*c>;
// generators/relations