metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.8Dic6, M4(2).30D6, C12.37(C4⋊C4), (C2×C12).28Q8, C12.441(C2×D4), (C2×C12).483D4, (C2×C4).16Dic6, C4.Dic3.6C4, (C22×C6).16Q8, C12.68(C22×C4), (C22×C4).150D6, C12.53D4⋊13C2, C22.4(C2×Dic6), C3⋊2(M4(2).C4), C4.21(Dic3⋊C4), (C2×C12).415C23, (C6×M4(2)).27C2, (C2×M4(2)).16S3, C4.Dic3.41C22, C22.17(Dic3⋊C4), (C22×C12).183C22, (C3×M4(2)).33C22, C3⋊C8.5(C2×C4), C4.90(S3×C2×C4), C6.52(C2×C4⋊C4), (C2×C4).49(C4×S3), (C2×C6).11(C2×Q8), (C2×C6).54(C4⋊C4), C4.131(C2×C3⋊D4), (C2×C12).103(C2×C4), (C2×C3⋊C8).143C22, C2.19(C2×Dic3⋊C4), (C2×C4).194(C3⋊D4), (C2×C4).511(C22×S3), (C2×C4.Dic3).24C2, SmallGroup(192,683)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C2×C4.Dic3 — C23.8Dic6 |
Generators and relations for C23.8Dic6
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=bd6, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd11 >
Subgroups: 184 in 102 conjugacy classes, 59 normal (39 characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22 [×3], C22, C6, C6 [×3], C8 [×8], C2×C4 [×6], C23, C12 [×4], C2×C6 [×3], C2×C6, C2×C8 [×4], M4(2) [×2], M4(2) [×8], C22×C4, C3⋊C8 [×4], C3⋊C8 [×2], C24 [×2], C2×C12 [×6], C22×C6, C8.C4 [×4], C2×M4(2), C2×M4(2) [×2], C2×C3⋊C8 [×2], C2×C3⋊C8, C4.Dic3 [×6], C4.Dic3, C2×C24, C3×M4(2) [×2], C3×M4(2), C22×C12, M4(2).C4, C12.53D4 [×4], C2×C4.Dic3 [×2], C6×M4(2), C23.8Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, M4(2).C4, C2×Dic3⋊C4, C23.8Dic6
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(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)
(1 31 7 37 13 43 19 25)(2 42 20 36 14 30 8 48)(3 41 9 47 15 29 21 35)(4 28 22 46 16 40 10 34)(5 27 11 33 17 39 23 45)(6 38 24 32 18 26 12 44)
G:=sub<Sym(48)| (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,31,7,37,13,43,19,25)(2,42,20,36,14,30,8,48)(3,41,9,47,15,29,21,35)(4,28,22,46,16,40,10,34)(5,27,11,33,17,39,23,45)(6,38,24,32,18,26,12,44)>;
G:=Group( (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,31,7,37,13,43,19,25)(2,42,20,36,14,30,8,48)(3,41,9,47,15,29,21,35)(4,28,22,46,16,40,10,34)(5,27,11,33,17,39,23,45)(6,38,24,32,18,26,12,44) );
G=PermutationGroup([(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(2,14),(4,16),(6,18),(8,20),(10,22),(12,24),(26,38),(28,40),(30,42),(32,44),(34,46),(36,48)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,31,7,37,13,43,19,25),(2,42,20,36,14,30,8,48),(3,41,9,47,15,29,21,35),(4,28,22,46,16,40,10,34),(5,27,11,33,17,39,23,45),(6,38,24,32,18,26,12,44)])
42 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | - | - | + | + | - | - | |||||
image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Q8 | D6 | D6 | Dic6 | C4×S3 | C3⋊D4 | Dic6 | M4(2).C4 | C23.8Dic6 |
kernel | C23.8Dic6 | C12.53D4 | C2×C4.Dic3 | C6×M4(2) | C4.Dic3 | C2×M4(2) | C2×C12 | C2×C12 | C22×C6 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C3 | C1 |
# reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 |
Matrix representation of C23.8Dic6 ►in GL6(𝔽73)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 72 | 0 |
0 | 0 | 72 | 0 | 0 | 72 |
72 | 0 | 0 | 0 | 0 | 0 |
0 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 72 | 72 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 72 | 0 | 0 | 72 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 72 | 0 | 0 |
0 | 0 | 0 | 0 | 72 | 0 |
0 | 0 | 0 | 0 | 0 | 72 |
64 | 0 | 0 | 0 | 0 | 0 |
13 | 65 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 71 | 0 | 0 |
0 | 0 | 60 | 1 | 0 | 0 |
0 | 0 | 59 | 72 | 0 | 46 |
0 | 0 | 1 | 1 | 72 | 0 |
33 | 25 | 0 | 0 | 0 | 0 |
44 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 71 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 0 | 14 | 0 | 72 | 0 |
0 | 0 | 13 | 27 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,72,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[64,13,0,0,0,0,0,65,0,0,0,0,0,0,72,60,59,1,0,0,71,1,72,1,0,0,0,0,0,72,0,0,0,0,46,0],[33,44,0,0,0,0,25,40,0,0,0,0,0,0,1,0,14,13,0,0,0,0,0,27,0,0,71,1,72,1,0,0,0,1,0,0] >;
C23.8Dic6 in GAP, Magma, Sage, TeX
C_2^3._8{\rm Dic}_6
% in TeX
G:=Group("C2^3.8Dic6");
// GroupNames label
G:=SmallGroup(192,683);
// by ID
G=gap.SmallGroup(192,683);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,422,58,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=b*d^6,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^11>;
// generators/relations