p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊23D4, C24.352C23, C23.501C24, C22.2062- 1+4, C22.2822+ 1+4, C23.Q8⋊34C2, C23.8Q8⋊78C2, C23.161(C4○D4), (C23×C4).412C22, (C2×C42).588C22, (C22×C4).552C23, C22.331(C22×D4), C24.C22⋊99C2, C23.10D4.29C2, C23.23D4.43C2, (C22×D4).184C22, C23.81C23⋊52C2, C23.65C23⋊97C2, C2.74(C22.19C24), C24.3C22.55C2, C23.63C23⋊105C2, C2.67(C22.45C24), C2.C42.231C22, C2.44(C22.26C24), C2.51(C22.50C24), C2.74(C22.46C24), C2.77(C22.47C24), C2.19(C22.31C24), (C4×C4⋊C4)⋊111C2, (C2×C4).369(C2×D4), (C2×C42⋊2C2)⋊13C2, (C2×C42⋊C2)⋊33C2, (C2×C4).409(C4○D4), (C2×C4⋊C4).341C22, C22.377(C2×C4○D4), (C2×C22⋊C4).515C22, SmallGroup(128,1333)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊23D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 484 in 259 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊2C2, C23×C4, C22×D4, C4×C4⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C24.3C22, C23.10D4, C23.Q8, C23.81C23, C2×C42⋊C2, C2×C42⋊2C2, C42⋊23D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C22.26C24, C22.31C24, C22.45C24, C22.46C24, C22.47C24, C22.50C24, C42⋊23D4
►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 57 25 17)(2 58 26 18)(3 59 27 19)(4 60 28 20)(5 33 9 50)(6 34 10 51)(7 35 11 52)(8 36 12 49)(13 29 53 47)(14 30 54 48)(15 31 55 45)(16 32 56 46)(21 38 61 43)(22 39 62 44)(23 40 63 41)(24 37 64 42)
(1 51 13 21)(2 50 14 24)(3 49 15 23)(4 52 16 22)(5 32 37 60)(6 31 38 59)(7 30 39 58)(8 29 40 57)(9 46 42 20)(10 45 43 19)(11 48 44 18)(12 47 41 17)(25 34 53 61)(26 33 54 64)(27 36 55 63)(28 35 56 62)
(2 28)(4 26)(5 42)(6 40)(7 44)(8 38)(9 37)(10 41)(11 39)(12 43)(14 56)(16 54)(17 19)(18 58)(20 60)(21 51)(22 33)(23 49)(24 35)(29 31)(30 48)(32 46)(34 61)(36 63)(45 47)(50 62)(52 64)(57 59)
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), (1,57,25,17)(2,58,26,18)(3,59,27,19)(4,60,28,20)(5,33,9,50)(6,34,10,51)(7,35,11,52)(8,36,12,49)(13,29,53,47)(14,30,54,48)(15,31,55,45)(16,32,56,46)(21,38,61,43)(22,39,62,44)(23,40,63,41)(24,37,64,42), (1,51,13,21)(2,50,14,24)(3,49,15,23)(4,52,16,22)(5,32,37,60)(6,31,38,59)(7,30,39,58)(8,29,40,57)(9,46,42,20)(10,45,43,19)(11,48,44,18)(12,47,41,17)(25,34,53,61)(26,33,54,64)(27,36,55,63)(28,35,56,62), (2,28)(4,26)(5,42)(6,40)(7,44)(8,38)(9,37)(10,41)(11,39)(12,43)(14,56)(16,54)(17,19)(18,58)(20,60)(21,51)(22,33)(23,49)(24,35)(29,31)(30,48)(32,46)(34,61)(36,63)(45,47)(50,62)(52,64)(57,59)>;
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), (1,57,25,17)(2,58,26,18)(3,59,27,19)(4,60,28,20)(5,33,9,50)(6,34,10,51)(7,35,11,52)(8,36,12,49)(13,29,53,47)(14,30,54,48)(15,31,55,45)(16,32,56,46)(21,38,61,43)(22,39,62,44)(23,40,63,41)(24,37,64,42), (1,51,13,21)(2,50,14,24)(3,49,15,23)(4,52,16,22)(5,32,37,60)(6,31,38,59)(7,30,39,58)(8,29,40,57)(9,46,42,20)(10,45,43,19)(11,48,44,18)(12,47,41,17)(25,34,53,61)(26,33,54,64)(27,36,55,63)(28,35,56,62), (2,28)(4,26)(5,42)(6,40)(7,44)(8,38)(9,37)(10,41)(11,39)(12,43)(14,56)(16,54)(17,19)(18,58)(20,60)(21,51)(22,33)(23,49)(24,35)(29,31)(30,48)(32,46)(34,61)(36,63)(45,47)(50,62)(52,64)(57,59) );
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)], [(1,57,25,17),(2,58,26,18),(3,59,27,19),(4,60,28,20),(5,33,9,50),(6,34,10,51),(7,35,11,52),(8,36,12,49),(13,29,53,47),(14,30,54,48),(15,31,55,45),(16,32,56,46),(21,38,61,43),(22,39,62,44),(23,40,63,41),(24,37,64,42)], [(1,51,13,21),(2,50,14,24),(3,49,15,23),(4,52,16,22),(5,32,37,60),(6,31,38,59),(7,30,39,58),(8,29,40,57),(9,46,42,20),(10,45,43,19),(11,48,44,18),(12,47,41,17),(25,34,53,61),(26,33,54,64),(27,36,55,63),(28,35,56,62)], [(2,28),(4,26),(5,42),(6,40),(7,44),(8,38),(9,37),(10,41),(11,39),(12,43),(14,56),(16,54),(17,19),(18,58),(20,60),(21,51),(22,33),(23,49),(24,35),(29,31),(30,48),(32,46),(34,61),(36,63),(45,47),(50,62),(52,64),(57,59)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊23D4 | C4×C4⋊C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C23.65C23 | C24.3C22 | C23.10D4 | C23.Q8 | C23.81C23 | C2×C42⋊C2 | C2×C42⋊2C2 | C42 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 12 | 4 | 1 | 1 |
Matrix representation of C42⋊23D4 ►in GL6(𝔽5)
| 3 | 2 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [3,1,0,0,0,0,2,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,2,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;
C42⋊23D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{23}D_4 % in TeX
G:=Group("C4^2:23D4"); // GroupNames label
G:=SmallGroup(128,1333);
// by ID
G=gap.SmallGroup(128,1333);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations