p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊C4.7C8, C4⋊C16⋊14C2, C4.19(C4⋊C8), C8.38(C4⋊C4), (C2×C8).62Q8, C8.42(C2×Q8), C8.135(C2×D4), (C2×C8).392D4, C22⋊C4.3C8, C8⋊C4.17C4, C2.5(D4○C16), (C22×C16).9C2, C23.21(C2×C8), C22.13(C4⋊C8), (C2×C16).65C22, C42.169(C2×C4), (C2×C8).628C23, (C4×C8).321C22, (C2×C4).51M4(2), C4.66(C2×M4(2)), C42⋊C2.21C4, (C2×M5(2)).24C2, (C2×M4(2)).33C4, C22.49(C22×C8), C8○2M4(2).20C2, (C22×C8).579C22, C4.80(C2×C4⋊C4), C2.14(C2×C4⋊C8), (C2×C4).26(C2×C8), (C2×C8).195(C2×C4), (C2×C4).138(C4⋊C4), (C2×C4).613(C22×C4), (C22×C4).409(C2×C4), SmallGroup(128,883)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4.7C8
G = < a,b,c | a4=b4=1, c8=a2, bab-1=a-1, ac=ca, cbc-1=a2b-1 >
Subgroups: 100 in 80 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, C2×C16, M5(2), C42⋊C2, C22×C8, C2×M4(2), C4⋊C16, C8○2M4(2), C22×C16, C2×M5(2), C4⋊C4.7C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, D4○C16, C4⋊C4.7C8
►On 64 points
(1 25 9 17)(2 26 10 18)(3 27 11 19)(4 28 12 20)(5 29 13 21)(6 30 14 22)(7 31 15 23)(8 32 16 24)(33 52 41 60)(34 53 42 61)(35 54 43 62)(36 55 44 63)(37 56 45 64)(38 57 46 49)(39 58 47 50)(40 59 48 51)
(1 56 21 33)(2 42 22 49)(3 58 23 35)(4 44 24 51)(5 60 25 37)(6 46 26 53)(7 62 27 39)(8 48 28 55)(9 64 29 41)(10 34 30 57)(11 50 31 43)(12 36 32 59)(13 52 17 45)(14 38 18 61)(15 54 19 47)(16 40 20 63)
(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)| (1,25,9,17)(2,26,10,18)(3,27,11,19)(4,28,12,20)(5,29,13,21)(6,30,14,22)(7,31,15,23)(8,32,16,24)(33,52,41,60)(34,53,42,61)(35,54,43,62)(36,55,44,63)(37,56,45,64)(38,57,46,49)(39,58,47,50)(40,59,48,51), (1,56,21,33)(2,42,22,49)(3,58,23,35)(4,44,24,51)(5,60,25,37)(6,46,26,53)(7,62,27,39)(8,48,28,55)(9,64,29,41)(10,34,30,57)(11,50,31,43)(12,36,32,59)(13,52,17,45)(14,38,18,61)(15,54,19,47)(16,40,20,63), (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( (1,25,9,17)(2,26,10,18)(3,27,11,19)(4,28,12,20)(5,29,13,21)(6,30,14,22)(7,31,15,23)(8,32,16,24)(33,52,41,60)(34,53,42,61)(35,54,43,62)(36,55,44,63)(37,56,45,64)(38,57,46,49)(39,58,47,50)(40,59,48,51), (1,56,21,33)(2,42,22,49)(3,58,23,35)(4,44,24,51)(5,60,25,37)(6,46,26,53)(7,62,27,39)(8,48,28,55)(9,64,29,41)(10,34,30,57)(11,50,31,43)(12,36,32,59)(13,52,17,45)(14,38,18,61)(15,54,19,47)(16,40,20,63), (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([[(1,25,9,17),(2,26,10,18),(3,27,11,19),(4,28,12,20),(5,29,13,21),(6,30,14,22),(7,31,15,23),(8,32,16,24),(33,52,41,60),(34,53,42,61),(35,54,43,62),(36,55,44,63),(37,56,45,64),(38,57,46,49),(39,58,47,50),(40,59,48,51)], [(1,56,21,33),(2,42,22,49),(3,58,23,35),(4,44,24,51),(5,60,25,37),(6,46,26,53),(7,62,27,39),(8,48,28,55),(9,64,29,41),(10,34,30,57),(11,50,31,43),(12,36,32,59),(13,52,17,45),(14,38,18,61),(15,54,19,47),(16,40,20,63)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D4 | Q8 | M4(2) | D4○C16 |
| kernel | C4⋊C4.7C8 | C4⋊C16 | C8○2M4(2) | C22×C16 | C2×M5(2) | C8⋊C4 | C42⋊C2 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×C8 | C2×C8 | C2×C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 16 |
Matrix representation of C4⋊C4.7C8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 16 | 13 |
| 16 | 16 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 1 | 8 |
| 0 | 0 | 4 | 16 |
| 16 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,16,0,0,0,13],[16,2,0,0,16,1,0,0,0,0,1,4,0,0,8,16],[16,0,0,0,16,1,0,0,0,0,12,0,0,0,0,12] >;
C4⋊C4.7C8 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._7C_8
% in TeX
G:=Group("C4:C4.7C8"); // GroupNames label
G:=SmallGroup(128,883);
// by ID
G=gap.SmallGroup(128,883);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,723,102,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^8=a^2,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations