p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.6M4(2), (C2×D4).1C8, (C2×Q8).1C8, C22⋊C16⋊2C2, (C2×C8).291D4, C2.5(C23⋊C8), C2.4(D4.C8), C4.36(C23⋊C4), (C2×M4(2)).5C4, (C22×C8).2C22, C4.21(C4.D4), C22.48(C22⋊C8), (C2×C4).10(C2×C8), (C2×C4○D4).1C4, (C22×C8)⋊C2.8C2, (C22×C4).164(C2×C4), (C2×C4).377(C22⋊C4), SmallGroup(128,47)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.M4(2)
G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, ab=ba, eae=ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=abcd5 >
Subgroups: 136 in 58 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C2×C16, C22×C8, C2×M4(2), C2×C4○D4, C22⋊C16, (C22×C8)⋊C2, C23.M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.D4, C23⋊C8, D4.C8, C23.M4(2)
►On 64 points
Generators in S64
(1 22)(2 59)(3 24)(4 61)(5 26)(6 63)(7 28)(8 49)(9 30)(10 51)(11 32)(12 53)(13 18)(14 55)(15 20)(16 57)(17 42)(19 44)(21 46)(23 48)(25 34)(27 36)(29 38)(31 40)(33 60)(35 62)(37 64)(39 50)(41 52)(43 54)(45 56)(47 58)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 61)(18 62)(19 63)(20 64)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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)
(2 63)(3 41)(4 21)(6 51)(7 45)(8 25)(10 55)(11 33)(12 29)(14 59)(15 37)(16 17)(18 26)(19 40)(20 56)(22 30)(23 44)(24 60)(27 48)(28 64)(31 36)(32 52)(34 57)(38 61)(42 49)(46 53)(50 58)(54 62)
G:=sub<Sym(64)| (1,22)(2,59)(3,24)(4,61)(5,26)(6,63)(7,28)(8,49)(9,30)(10,51)(11,32)(12,53)(13,18)(14,55)(15,20)(16,57)(17,42)(19,44)(21,46)(23,48)(25,34)(27,36)(29,38)(31,40)(33,60)(35,62)(37,64)(39,50)(41,52)(43,54)(45,56)(47,58), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,61)(18,62)(19,63)(20,64)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (2,63)(3,41)(4,21)(6,51)(7,45)(8,25)(10,55)(11,33)(12,29)(14,59)(15,37)(16,17)(18,26)(19,40)(20,56)(22,30)(23,44)(24,60)(27,48)(28,64)(31,36)(32,52)(34,57)(38,61)(42,49)(46,53)(50,58)(54,62)>;
G:=Group( (1,22)(2,59)(3,24)(4,61)(5,26)(6,63)(7,28)(8,49)(9,30)(10,51)(11,32)(12,53)(13,18)(14,55)(15,20)(16,57)(17,42)(19,44)(21,46)(23,48)(25,34)(27,36)(29,38)(31,40)(33,60)(35,62)(37,64)(39,50)(41,52)(43,54)(45,56)(47,58), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,61)(18,62)(19,63)(20,64)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (2,63)(3,41)(4,21)(6,51)(7,45)(8,25)(10,55)(11,33)(12,29)(14,59)(15,37)(16,17)(18,26)(19,40)(20,56)(22,30)(23,44)(24,60)(27,48)(28,64)(31,36)(32,52)(34,57)(38,61)(42,49)(46,53)(50,58)(54,62) );
G=PermutationGroup([[(1,22),(2,59),(3,24),(4,61),(5,26),(6,63),(7,28),(8,49),(9,30),(10,51),(11,32),(12,53),(13,18),(14,55),(15,20),(16,57),(17,42),(19,44),(21,46),(23,48),(25,34),(27,36),(29,38),(31,40),(33,60),(35,62),(37,64),(39,50),(41,52),(43,54),(45,56),(47,58)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,61),(18,62),(19,63),(20,64),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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)], [(2,63),(3,41),(4,21),(6,51),(7,45),(8,25),(10,55),(11,33),(12,29),(14,59),(15,37),(16,17),(18,26),(19,40),(20,56),(22,30),(23,44),(24,60),(27,48),(28,64),(31,36),(32,52),(34,57),(38,61),(42,49),(46,53),(50,58),(54,62)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 8I | 8J | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 8 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | M4(2) | D4.C8 | C23⋊C4 | C4.D4 |
| kernel | C23.M4(2) | C22⋊C16 | (C22×C8)⋊C2 | C2×M4(2) | C2×C4○D4 | C2×D4 | C2×Q8 | C2×C8 | C23 | C2 | C4 | C4 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 16 | 1 | 1 |
Matrix representation of C23.M4(2) ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 13 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 8 | 9 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 3 | 16 |
| 0 | 0 | 7 | 14 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 16 |
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,16,0,0,0,13,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[8,8,0,0,9,8,0,0,0,0,3,7,0,0,16,14],[1,0,0,0,0,16,0,0,0,0,1,8,0,0,0,16] >;
C23.M4(2) in GAP, Magma, Sage, TeX
C_2^3.M_4(2)
% in TeX
G:=Group("C2^3.M4(2)"); // GroupNames label
G:=SmallGroup(128,47);
// by ID
G=gap.SmallGroup(128,47);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,723,352,1242,136,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^8=c,a*b=b*a,e*a*e=a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a*b*c*d^5>;
// generators/relations