p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.51(C4×D4), (C2×Q8).3Q8, C4⋊C4.301D4, C4.10D4⋊2C4, (C22×C4).60D4, C42⋊6C4.1C2, C23.553(C2×D4), M4(2).1(C2×C4), C4.24(C22⋊Q8), C22.81C22≀C2, C2.2(D4.8D4), C2.2(D4.10D4), M4(2)⋊C4.1C2, C42⋊C2.9C22, (C22×C4).673C23, (C2×C42).264C22, (C2×M4(2)).6C22, (C22×Q8).10C22, C2.13(C23.8Q8), C23.32C23.2C2, C23.67C23.4C2, C22.46(C22.D4), (C2×C4).8(C2×Q8), (C2×C4).5(C4⋊C4), (C2×C4).983(C2×D4), C22.23(C2×C4⋊C4), (C2×C4).7(C22×C4), (C2×Q8).59(C2×C4), (C2×C4⋊C4).55C22, (C2×C4).316(C4○D4), (C2×C4.10D4).1C2, SmallGroup(128,589)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.10D4⋊2C4
G = < a,b,c,d | a4=d4=1, b4=a2, c2=bab-1=a-1, ac=ca, ad=da, cbc-1=a-1b3, dbd-1=a2b3, dcd-1=a2b2c >
Subgroups: 244 in 136 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C4.10D4, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×Q8, C2×M4(2), C22×Q8, C42⋊6C4, C23.67C23, C2×C4.10D4, M4(2)⋊C4, C23.32C23, C4.10D4⋊2C4
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, D4.8D4, D4.10D4, C4.10D4⋊2C4
►On 32 points
(1 25 5 29)(2 30 6 26)(3 27 7 31)(4 32 8 28)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)
(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 18 29 10 5 22 25 14)(2 9 26 17 6 13 30 21)(3 24 31 16 7 20 27 12)(4 15 28 23 8 11 32 19)
(1 22 31 12)(2 21 32 11)(3 20 25 10)(4 19 26 9)(5 18 27 16)(6 17 28 15)(7 24 29 14)(8 23 30 13)
G:=sub<Sym(32)| (1,25,5,29)(2,30,6,26)(3,27,7,31)(4,32,8,28)(9,21,13,17)(10,18,14,22)(11,23,15,19)(12,20,16,24), (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), (1,18,29,10,5,22,25,14)(2,9,26,17,6,13,30,21)(3,24,31,16,7,20,27,12)(4,15,28,23,8,11,32,19), (1,22,31,12)(2,21,32,11)(3,20,25,10)(4,19,26,9)(5,18,27,16)(6,17,28,15)(7,24,29,14)(8,23,30,13)>;
G:=Group( (1,25,5,29)(2,30,6,26)(3,27,7,31)(4,32,8,28)(9,21,13,17)(10,18,14,22)(11,23,15,19)(12,20,16,24), (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), (1,18,29,10,5,22,25,14)(2,9,26,17,6,13,30,21)(3,24,31,16,7,20,27,12)(4,15,28,23,8,11,32,19), (1,22,31,12)(2,21,32,11)(3,20,25,10)(4,19,26,9)(5,18,27,16)(6,17,28,15)(7,24,29,14)(8,23,30,13) );
G=PermutationGroup([[(1,25,5,29),(2,30,6,26),(3,27,7,31),(4,32,8,28),(9,21,13,17),(10,18,14,22),(11,23,15,19),(12,20,16,24)], [(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)], [(1,18,29,10,5,22,25,14),(2,9,26,17,6,13,30,21),(3,24,31,16,7,20,27,12),(4,15,28,23,8,11,32,19)], [(1,22,31,12),(2,21,32,11),(3,20,25,10),(4,19,26,9),(5,18,27,16),(6,17,28,15),(7,24,29,14),(8,23,30,13)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4T | 4U | 4V | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | C4○D4 | D4.8D4 | D4.10D4 |
| kernel | C4.10D4⋊2C4 | C42⋊6C4 | C23.67C23 | C2×C4.10D4 | M4(2)⋊C4 | C23.32C23 | C4.10D4 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C4.10D4⋊2C4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 1 | 0 | 16 | 4 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 13 | 16 | 8 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 7 | 13 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 1 | 13 | 16 | 8 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 6 | 16 | 4 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 1 | 13 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,1,0,0,0,4,0,0,0,0,0,0,13,16,0,0,0,0,0,4],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,11,0,0,13,0,1,7,0,0,16,13,0,13,0,0,8,0,0,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,1,1,7,0,0,0,13,0,6,0,0,4,16,0,16,0,0,0,8,0,4],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,4,16,4,0,0,0,1,0,1,0,0,1,13,0,0,0,0,0,15,0,16] >;
C4.10D4⋊2C4 in GAP, Magma, Sage, TeX
C_4._{10}D_4\rtimes_2C_4 % in TeX
G:=Group("C4.10D4:2C4"); // GroupNames label
G:=SmallGroup(128,589);
// by ID
G=gap.SmallGroup(128,589);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,352,2019,1018,521,248,1411]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^4=1,b^4=a^2,c^2=b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^-1*b^3,d*b*d^-1=a^2*b^3,d*c*d^-1=a^2*b^2*c>;
// generators/relations