p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).26D4, (C2×D4).9Q8, (C2×Q8).5Q8, C23⋊C4.2C4, C23.6(C4⋊C4), C22.34(C4×D4), (C22×C4).62D4, C4.120C22≀C2, Q8○M4(2).3C2, C4.98(C22⋊Q8), C23.6(C22×C4), (C22×C8).28C22, M4(2)⋊4C4⋊14C2, (C22×C4).678C23, C42⋊C2.14C22, C42.6C22⋊21C2, C23.C23.5C2, C2.24(C23.8Q8), C4.107(C22.D4), (C2×M4(2)).176C22, (C2×C4).7(C4⋊C4), (C2×C4).12(C2×Q8), (C2×D4).69(C2×C4), C22⋊C4.4(C2×C4), C22.26(C2×C4⋊C4), (C2×C4).53(C4○D4), (C2×C4).1327(C2×D4), (C2×C4○D4).14C22, (C22×C8)⋊C2.14C2, SmallGroup(128,600)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×D4).Q8
G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=d2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc=dbd-1=b-1, be=eb, cd=dc, ece-1=ab2c, ede-1=abd3 >
Subgroups: 228 in 130 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C23⋊C4, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, M4(2)⋊4C4, (C22×C8)⋊C2, C23.C23, C42.6C22, Q8○M4(2), (C2×D4).Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8, (C2×D4).Q8
►On 32 points
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(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)
(1 13 3 15 5 9 7 11)(2 23 4 17 6 19 8 21)(10 30 12 32 14 26 16 28)(18 29 20 31 22 25 24 27)
G:=sub<Sym(32)| (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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), (1,13,3,15,5,9,7,11)(2,23,4,17,6,19,8,21)(10,30,12,32,14,26,16,28)(18,29,20,31,22,25,24,27)>;
G:=Group( (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (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), (1,13,3,15,5,9,7,11)(2,23,4,17,6,19,8,21)(10,30,12,32,14,26,16,28)(18,29,20,31,22,25,24,27) );
G=PermutationGroup([[(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17)], [(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12),(17,23,21,19),(18,20,22,24),(25,27,29,31),(26,32,30,28)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(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)], [(1,13,3,15,5,9,7,11),(2,23,4,17,6,19,8,21),(10,30,12,32,14,26,16,28),(18,29,20,31,22,25,24,27)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | ··· | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | Q8 | C4○D4 | (C2×D4).Q8 |
| kernel | (C2×D4).Q8 | M4(2)⋊4C4 | (C22×C8)⋊C2 | C23.C23 | C42.6C22 | Q8○M4(2) | C23⋊C4 | C2×C8 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 4 | 4 |
Matrix representation of (C2×D4).Q8 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 16 | 16 | 1 | 2 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 4 | 4 | 0 | 13 |
| 0 | 0 | 1 | 0 |
| 1 | 1 | 16 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 8 | 8 | 9 | 1 |
| 0 | 0 | 9 | 0 |
| 0 | 8 | 0 | 0 |
| 8 | 0 | 9 | 9 |
| 0 | 15 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(17))| [0,16,16,0,16,0,16,0,0,0,1,0,0,0,2,16],[4,0,0,4,0,4,0,4,0,0,13,0,0,0,0,13],[0,1,1,0,0,1,0,0,1,16,0,0,0,15,0,16],[8,0,0,8,8,0,8,0,9,9,0,9,1,0,0,9],[0,15,0,0,15,0,0,0,0,0,15,0,0,0,0,15] >;
(C2×D4).Q8 in GAP, Magma, Sage, TeX
(C_2\times D_4).Q_8
% in TeX
G:=Group("(C2xD4).Q8"); // GroupNames label
G:=SmallGroup(128,600);
// by ID
G=gap.SmallGroup(128,600);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,521,1411,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=b^2,e^2=d^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=a*b^2*c,e*d*e^-1=a*b*d^3>;
// generators/relations