p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.23C23, C4○D4⋊2Q8, D4.7(C2×Q8), Q8.7(C2×Q8), C4⋊C4.344D4, D4⋊2Q8⋊4C2, Q8⋊Q8⋊4C2, D4.Q8⋊16C2, Q8.Q8⋊16C2, C4⋊C8.47C22, C4⋊C4.47C23, (C2×C8).31C23, C4.35(C22×Q8), (C2×C4).282C24, C22⋊C4.145D4, (C4×D4).71C22, C23.451(C2×D4), C4⋊Q8.104C22, C4.20(C22⋊Q8), (C4×Q8).68C22, C2.D8.82C22, C2.22(D4○SD16), (C2×D4).399C23, (C2×Q8).370C23, M4(2)⋊C4⋊21C2, C4.Q8.149C22, D4⋊C4.28C22, (C22×C8).345C22, Q8⋊C4.29C22, C23.24D4.8C2, C23.36D4.4C2, C22.542(C22×D4), C22.21(C22⋊Q8), C42.C2.11C22, C23.41C23⋊5C2, (C22×C4).1001C23, C42.6C22⋊11C2, (C2×M4(2)).71C22, C42⋊C2.121C22, C23.33C23.8C2, (C2×C4.Q8)⋊30C2, C4.92(C2×C4○D4), (C2×C4).484(C2×D4), (C2×C4).106(C2×Q8), C2.63(C2×C22⋊Q8), (C2×C4).484(C4○D4), (C2×C4⋊C4).608C22, (C2×C4○D4).135C22, SmallGroup(128,1816)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.23C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=a2, ab=ba, cac-1=a-1, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >
Subgroups: 332 in 188 conjugacy classes, 100 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C2×C4○D4, C23.24D4, C23.36D4, C42.6C22, C2×C4.Q8, M4(2)⋊C4, Q8⋊Q8, D4⋊2Q8, D4.Q8, Q8.Q8, C23.33C23, C23.41C23, C42.23C23
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D4○SD16, C42.23C23
►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 42 46 17)(2 43 47 18)(3 44 48 19)(4 41 45 20)(5 62 33 39)(6 63 34 40)(7 64 35 37)(8 61 36 38)(9 15 23 49)(10 16 24 50)(11 13 21 51)(12 14 22 52)(25 56 60 31)(26 53 57 32)(27 54 58 29)(28 55 59 30)
(1 57 48 28)(2 60 45 27)(3 59 46 26)(4 58 47 25)(5 52 35 16)(6 51 36 15)(7 50 33 14)(8 49 34 13)(9 63 21 38)(10 62 22 37)(11 61 23 40)(12 64 24 39)(17 32 44 55)(18 31 41 54)(19 30 42 53)(20 29 43 56)
(5 62)(6 63)(7 64)(8 61)(13 51)(14 52)(15 49)(16 50)(17 42)(18 43)(19 44)(20 41)(25 31)(26 32)(27 29)(28 30)(33 39)(34 40)(35 37)(36 38)(53 57)(54 58)(55 59)(56 60)
(1 22 3 24)(2 9 4 11)(5 30 7 32)(6 56 8 54)(10 46 12 48)(13 43 15 41)(14 19 16 17)(18 49 20 51)(21 47 23 45)(25 38 27 40)(26 62 28 64)(29 34 31 36)(33 55 35 53)(37 57 39 59)(42 52 44 50)(58 63 60 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), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,57,48,28)(2,60,45,27)(3,59,46,26)(4,58,47,25)(5,52,35,16)(6,51,36,15)(7,50,33,14)(8,49,34,13)(9,63,21,38)(10,62,22,37)(11,61,23,40)(12,64,24,39)(17,32,44,55)(18,31,41,54)(19,30,42,53)(20,29,43,56), (5,62)(6,63)(7,64)(8,61)(13,51)(14,52)(15,49)(16,50)(17,42)(18,43)(19,44)(20,41)(25,31)(26,32)(27,29)(28,30)(33,39)(34,40)(35,37)(36,38)(53,57)(54,58)(55,59)(56,60), (1,22,3,24)(2,9,4,11)(5,30,7,32)(6,56,8,54)(10,46,12,48)(13,43,15,41)(14,19,16,17)(18,49,20,51)(21,47,23,45)(25,38,27,40)(26,62,28,64)(29,34,31,36)(33,55,35,53)(37,57,39,59)(42,52,44,50)(58,63,60,61)>;
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), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,57,48,28)(2,60,45,27)(3,59,46,26)(4,58,47,25)(5,52,35,16)(6,51,36,15)(7,50,33,14)(8,49,34,13)(9,63,21,38)(10,62,22,37)(11,61,23,40)(12,64,24,39)(17,32,44,55)(18,31,41,54)(19,30,42,53)(20,29,43,56), (5,62)(6,63)(7,64)(8,61)(13,51)(14,52)(15,49)(16,50)(17,42)(18,43)(19,44)(20,41)(25,31)(26,32)(27,29)(28,30)(33,39)(34,40)(35,37)(36,38)(53,57)(54,58)(55,59)(56,60), (1,22,3,24)(2,9,4,11)(5,30,7,32)(6,56,8,54)(10,46,12,48)(13,43,15,41)(14,19,16,17)(18,49,20,51)(21,47,23,45)(25,38,27,40)(26,62,28,64)(29,34,31,36)(33,55,35,53)(37,57,39,59)(42,52,44,50)(58,63,60,61) );
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)], [(1,42,46,17),(2,43,47,18),(3,44,48,19),(4,41,45,20),(5,62,33,39),(6,63,34,40),(7,64,35,37),(8,61,36,38),(9,15,23,49),(10,16,24,50),(11,13,21,51),(12,14,22,52),(25,56,60,31),(26,53,57,32),(27,54,58,29),(28,55,59,30)], [(1,57,48,28),(2,60,45,27),(3,59,46,26),(4,58,47,25),(5,52,35,16),(6,51,36,15),(7,50,33,14),(8,49,34,13),(9,63,21,38),(10,62,22,37),(11,61,23,40),(12,64,24,39),(17,32,44,55),(18,31,41,54),(19,30,42,53),(20,29,43,56)], [(5,62),(6,63),(7,64),(8,61),(13,51),(14,52),(15,49),(16,50),(17,42),(18,43),(19,44),(20,41),(25,31),(26,32),(27,29),(28,30),(33,39),(34,40),(35,37),(36,38),(53,57),(54,58),(55,59),(56,60)], [(1,22,3,24),(2,9,4,11),(5,30,7,32),(6,56,8,54),(10,46,12,48),(13,43,15,41),(14,19,16,17),(18,49,20,51),(21,47,23,45),(25,38,27,40),(26,62,28,64),(29,34,31,36),(33,55,35,53),(37,57,39,59),(42,52,44,50),(58,63,60,61)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | D4○SD16 |
| kernel | C42.23C23 | C23.24D4 | C23.36D4 | C42.6C22 | C2×C4.Q8 | M4(2)⋊C4 | Q8⋊Q8 | D4⋊2Q8 | D4.Q8 | Q8.Q8 | C23.33C23 | C23.41C23 | C22⋊C4 | C4⋊C4 | C4○D4 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42.23C23 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 1 | 0 |
| 0 | 0 | 16 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 | 5 |
| 0 | 0 | 12 | 12 | 5 | 5 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 1 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,16,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,12,0,12,0,0,7,0,12,12,0,0,0,0,12,5,0,0,0,0,5,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,2,1,1,1,0,0,0,16,0,0] >;
C42.23C23 in GAP, Magma, Sage, TeX
C_4^2._{23}C_2^3 % in TeX
G:=Group("C4^2.23C2^3"); // GroupNames label
G:=SmallGroup(128,1816);
// by ID
G=gap.SmallGroup(128,1816);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,1018,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2*b^2,e^2=a^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*c,d*e=e*d>;
// generators/relations