p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.385C23, C8⋊3Q8⋊6C2, C8⋊8D4⋊45C2, C4⋊C4.245D4, C8.8(C4○D4), C8.D4⋊6C2, C8⋊2D4.4C2, D8⋊C4⋊13C2, (C4×SD16)⋊53C2, C22⋊C4.85D4, C23.82(C2×D4), Q16⋊C4⋊13C2, C8.12D4⋊18C2, C4⋊C4.112C23, (C4×C8).290C22, (C2×C8).600C23, (C2×C4).371C24, (C4×D4).92C22, (C2×D8).63C22, C4⋊Q8.114C22, (C4×Q8).89C22, C8○2M4(2)⋊15C2, C2.37(D4○SD16), (C2×D4).126C23, C4⋊D4.33C22, (C2×Q8).114C23, (C2×Q16).64C22, C4.Q8.163C22, C8⋊C4.128C22, C22⋊Q8.33C22, (C22×C8).359C22, C4.4D4.34C22, C22.631(C22×D4), D4⋊C4.204C22, (C22×C4).1051C23, C22.36C24⋊4C2, Q8⋊C4.206C22, (C2×SD16).151C22, C42.28C22⋊33C2, C42⋊C2.328C22, (C2×M4(2)).281C22, C2.68(C22.26C24), C4.56(C2×C4○D4), (C2×C4).143(C2×D4), SmallGroup(128,1905)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
C1 — C22 — C42⋊C2 — C42.385C23 |
Generators and relations for C42.385C23
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b, ab=ba, ac=ca, dad=ab2, ae=ea, cbc=b-1, bd=db, be=eb, dcd=a2c, ece-1=b-1c, de=ed >
Subgroups: 348 in 181 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C4.Q8, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C8○2M4(2), C4×SD16, Q16⋊C4, D8⋊C4, C8⋊8D4, C8⋊2D4, C8.D4, C42.28C22, C8.12D4, C8⋊3Q8, C22.36C24, C42.385C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.26C24, D4○SD16, C42.385C23
(1 39 31 16)(2 40 32 9)(3 33 25 10)(4 34 26 11)(5 35 27 12)(6 36 28 13)(7 37 29 14)(8 38 30 15)(17 57 46 55)(18 58 47 56)(19 59 48 49)(20 60 41 50)(21 61 42 51)(22 62 43 52)(23 63 44 53)(24 64 45 54)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 47)(2 42)(3 45)(4 48)(5 43)(6 46)(7 41)(8 44)(9 61)(10 64)(11 59)(12 62)(13 57)(14 60)(15 63)(16 58)(17 28)(18 31)(19 26)(20 29)(21 32)(22 27)(23 30)(24 25)(33 54)(34 49)(35 52)(36 55)(37 50)(38 53)(39 56)(40 51)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 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)
G:=sub<Sym(64)| (1,39,31,16)(2,40,32,9)(3,33,25,10)(4,34,26,11)(5,35,27,12)(6,36,28,13)(7,37,29,14)(8,38,30,15)(17,57,46,55)(18,58,47,56)(19,59,48,49)(20,60,41,50)(21,61,42,51)(22,62,43,52)(23,63,44,53)(24,64,45,54), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,47)(2,42)(3,45)(4,48)(5,43)(6,46)(7,41)(8,44)(9,61)(10,64)(11,59)(12,62)(13,57)(14,60)(15,63)(16,58)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(33,54)(34,49)(35,52)(36,55)(37,50)(38,53)(39,56)(40,51), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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)>;
G:=Group( (1,39,31,16)(2,40,32,9)(3,33,25,10)(4,34,26,11)(5,35,27,12)(6,36,28,13)(7,37,29,14)(8,38,30,15)(17,57,46,55)(18,58,47,56)(19,59,48,49)(20,60,41,50)(21,61,42,51)(22,62,43,52)(23,63,44,53)(24,64,45,54), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,47)(2,42)(3,45)(4,48)(5,43)(6,46)(7,41)(8,44)(9,61)(10,64)(11,59)(12,62)(13,57)(14,60)(15,63)(16,58)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(33,54)(34,49)(35,52)(36,55)(37,50)(38,53)(39,56)(40,51), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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) );
G=PermutationGroup([[(1,39,31,16),(2,40,32,9),(3,33,25,10),(4,34,26,11),(5,35,27,12),(6,36,28,13),(7,37,29,14),(8,38,30,15),(17,57,46,55),(18,58,47,56),(19,59,48,49),(20,60,41,50),(21,61,42,51),(22,62,43,52),(23,63,44,53),(24,64,45,54)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,47),(2,42),(3,45),(4,48),(5,43),(6,46),(7,41),(8,44),(9,61),(10,64),(11,59),(12,62),(13,57),(14,60),(15,63),(16,58),(17,28),(18,31),(19,26),(20,29),(21,32),(22,27),(23,30),(24,25),(33,54),(34,49),(35,52),(36,55),(37,50),(38,53),(39,56),(40,51)], [(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64)], [(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)]])
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D4○SD16 |
kernel | C42.385C23 | C8○2M4(2) | C4×SD16 | Q16⋊C4 | D8⋊C4 | C8⋊8D4 | C8⋊2D4 | C8.D4 | C42.28C22 | C8.12D4 | C8⋊3Q8 | C22.36C24 | C22⋊C4 | C4⋊C4 | C8 | C2 |
# reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 |
Matrix representation of C42.385C23 ►in GL6(𝔽17)
13 | 0 | 0 | 0 | 0 | 0 |
0 | 13 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 4 | 13 | 1 | 15 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 4 | 13 | 1 | 15 |
0 | 0 | 4 | 0 | 1 | 16 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 4 |
0 | 0 | 0 | 0 | 13 | 4 |
0 | 0 | 13 | 4 | 16 | 1 |
0 | 0 | 13 | 0 | 16 | 1 |
0 | 4 | 0 | 0 | 0 | 0 |
13 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 11 | 3 | 12 | 10 |
0 | 0 | 0 | 14 | 5 | 0 |
0 | 0 | 0 | 12 | 3 | 0 |
0 | 0 | 5 | 7 | 14 | 6 |
0 | 1 | 0 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 5 | 0 | 0 |
0 | 0 | 12 | 12 | 0 | 0 |
0 | 0 | 3 | 14 | 0 | 7 |
0 | 0 | 3 | 0 | 5 | 7 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,16,0,0,0,0,13,0,0,0,0,1,1,0,0,0,0,0,15,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,4,4,0,0,1,0,13,0,0,0,0,0,1,1,0,0,0,0,15,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,13,0,0,16,0,4,0,0,0,0,13,16,16,0,0,4,4,1,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,11,0,0,5,0,0,3,14,12,7,0,0,12,5,3,14,0,0,10,0,0,6],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,3,3,0,0,5,12,14,0,0,0,0,0,0,5,0,0,0,0,7,7] >;
C42.385C23 in GAP, Magma, Sage, TeX
C_4^2._{385}C_2^3
% in TeX
G:=Group("C4^2.385C2^3");
// GroupNames label
G:=SmallGroup(128,1905);
// by ID
G=gap.SmallGroup(128,1905);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,184,1018,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=b,a*b=b*a,a*c=c*a,d*a*d=a*b^2,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d=a^2*c,e*c*e^-1=b^-1*c,d*e=e*d>;
// generators/relations