p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.21M4(2), C22⋊C4⋊3C8, C2.10(C8×D4), C22⋊2(C4⋊C8), (C2×C8).319D4, (C23×C8).7C2, C2.4(C8⋊9D4), C24.95(C2×C4), C23.19(C2×C8), C22.98(C4×D4), C4.114C22≀C2, (C22×C4).79Q8, C23.70(C4⋊C4), (C22×C4).680D4, C22.29(C8○D4), C4.111(C22⋊Q8), C22.39(C22×C8), (C22×C8).25C22, C2.C42.19C4, (C23×C4).635C22, (C2×C42).261C22, C22.7C42⋊6C2, C23.268(C22×C4), C22.50(C2×M4(2)), C2.3(C23.8Q8), (C22×C4).1627C23, C4.131(C22.D4), C2.3(C42.6C22), (C2×C4)⋊2(C2×C8), (C2×C4⋊C8)⋊13C2, C2.11(C2×C4⋊C8), (C2×C4).90(C4⋊C4), C22.66(C2×C4⋊C4), (C2×C4).341(C2×Q8), (C2×C4).1525(C2×D4), (C2×C22⋊C8).20C2, (C4×C22⋊C4).13C2, (C2×C22⋊C4).28C4, (C2×C4).933(C4○D4), (C22×C4).116(C2×C4), SmallGroup(128,582)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.21M4(2)
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd5 >
Subgroups: 276 in 174 conjugacy classes, 80 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C22×C8, C22×C8, C22×C8, C23×C4, C22.7C42, C4×C22⋊C4, C2×C22⋊C8, C2×C4⋊C8, C23×C8, C23.21M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4⋊C8, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C22×C8, C2×M4(2), C8○D4, C23.8Q8, C2×C4⋊C8, C42.6C22, C8×D4, C8⋊9D4, C23.21M4(2)
►On 64 points
(1 41)(2 31)(3 43)(4 25)(5 45)(6 27)(7 47)(8 29)(9 60)(10 18)(11 62)(12 20)(13 64)(14 22)(15 58)(16 24)(17 37)(19 39)(21 33)(23 35)(26 53)(28 55)(30 49)(32 51)(34 57)(36 59)(38 61)(40 63)(42 50)(44 52)(46 54)(48 56)
(1 45)(2 46)(3 47)(4 48)(5 41)(6 42)(7 43)(8 44)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 56)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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 23 45 11)(2 16 46 20)(3 17 47 13)(4 10 48 22)(5 19 41 15)(6 12 42 24)(7 21 43 9)(8 14 44 18)(25 61 56 34)(26 39 49 58)(27 63 50 36)(28 33 51 60)(29 57 52 38)(30 35 53 62)(31 59 54 40)(32 37 55 64)
G:=sub<Sym(64)| (1,41)(2,31)(3,43)(4,25)(5,45)(6,27)(7,47)(8,29)(9,60)(10,18)(11,62)(12,20)(13,64)(14,22)(15,58)(16,24)(17,37)(19,39)(21,33)(23,35)(26,53)(28,55)(30,49)(32,51)(34,57)(36,59)(38,61)(40,63)(42,50)(44,52)(46,54)(48,56), (1,45)(2,46)(3,47)(4,48)(5,41)(6,42)(7,43)(8,44)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,23,45,11)(2,16,46,20)(3,17,47,13)(4,10,48,22)(5,19,41,15)(6,12,42,24)(7,21,43,9)(8,14,44,18)(25,61,56,34)(26,39,49,58)(27,63,50,36)(28,33,51,60)(29,57,52,38)(30,35,53,62)(31,59,54,40)(32,37,55,64)>;
G:=Group( (1,41)(2,31)(3,43)(4,25)(5,45)(6,27)(7,47)(8,29)(9,60)(10,18)(11,62)(12,20)(13,64)(14,22)(15,58)(16,24)(17,37)(19,39)(21,33)(23,35)(26,53)(28,55)(30,49)(32,51)(34,57)(36,59)(38,61)(40,63)(42,50)(44,52)(46,54)(48,56), (1,45)(2,46)(3,47)(4,48)(5,41)(6,42)(7,43)(8,44)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,23,45,11)(2,16,46,20)(3,17,47,13)(4,10,48,22)(5,19,41,15)(6,12,42,24)(7,21,43,9)(8,14,44,18)(25,61,56,34)(26,39,49,58)(27,63,50,36)(28,33,51,60)(29,57,52,38)(30,35,53,62)(31,59,54,40)(32,37,55,64) );
G=PermutationGroup([[(1,41),(2,31),(3,43),(4,25),(5,45),(6,27),(7,47),(8,29),(9,60),(10,18),(11,62),(12,20),(13,64),(14,22),(15,58),(16,24),(17,37),(19,39),(21,33),(23,35),(26,53),(28,55),(30,49),(32,51),(34,57),(36,59),(38,61),(40,63),(42,50),(44,52),(46,54),(48,56)], [(1,45),(2,46),(3,47),(4,48),(5,41),(6,42),(7,43),(8,44),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20),(25,56),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43)], [(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)], [(1,23,45,11),(2,16,46,20),(3,17,47,13),(4,10,48,22),(5,19,41,15),(6,12,42,24),(7,21,43,9),(8,14,44,18),(25,61,56,34),(26,39,49,58),(27,63,50,36),(28,33,51,60),(29,57,52,38),(30,35,53,62),(31,59,54,40),(32,37,55,64)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | Q8 | C4○D4 | M4(2) | C8○D4 |
| kernel | C23.21M4(2) | C22.7C42 | C4×C22⋊C4 | C2×C22⋊C8 | C2×C4⋊C8 | C23×C8 | C2.C42 | C2×C22⋊C4 | C22⋊C4 | C2×C8 | C22×C4 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 16 | 4 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C23.21M4(2) ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,13,0],[4,0,0,0,0,0,0,15,0,0,0,8,0,0,0,0,0,0,0,9,0,0,0,2,0] >;
C23.21M4(2) in GAP, Magma, Sage, TeX
C_2^3._{21}M_4(2) % in TeX
G:=Group("C2^3.21M4(2)"); // GroupNames label
G:=SmallGroup(128,582);
// by ID
G=gap.SmallGroup(128,582);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^5>;
// generators/relations