direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4.10D4, M4(2).7C22, C4.45(C2×D4), (C2×Q8).6C4, (C2×C4).121D4, (C22×C4).5C4, (C2×C4).2C23, C23.30(C2×C4), (C22×Q8).3C2, C4.11(C22⋊C4), C22.9(C22×C4), (C2×Q8).38C22, (C2×M4(2)).11C2, (C22×C4).32C22, C22.31(C22⋊C4), (C2×C4).6(C2×C4), C2.15(C2×C22⋊C4), SmallGroup(64,93)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4.10D4
G = < a,b,c,d | a2=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Subgroups: 105 in 73 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], Q8 [×8], C23, C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×4], C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C2×C4.10D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C2×C4.10D4
Character table of C2×C4.10D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | i | -i | i | -i | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | -i | -i | i | i | -i | i | i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | i | -i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -i | i | -i | i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | i | i | i | -i | -i | i | -i | -i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
►On 32 points
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 13 5 9)(2 10 6 14)(3 15 7 11)(4 12 8 16)(17 31 21 27)(18 28 22 32)(19 25 23 29)(20 30 24 26)
(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 20 9 26 5 24 13 30)(2 25 14 19 6 29 10 23)(3 18 11 32 7 22 15 28)(4 31 16 17 8 27 12 21)
G:=sub<Sym(32)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,13,5,9)(2,10,6,14)(3,15,7,11)(4,12,8,16)(17,31,21,27)(18,28,22,32)(19,25,23,29)(20,30,24,26), (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), (1,20,9,26,5,24,13,30)(2,25,14,19,6,29,10,23)(3,18,11,32,7,22,15,28)(4,31,16,17,8,27,12,21)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,13,5,9)(2,10,6,14)(3,15,7,11)(4,12,8,16)(17,31,21,27)(18,28,22,32)(19,25,23,29)(20,30,24,26), (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), (1,20,9,26,5,24,13,30)(2,25,14,19,6,29,10,23)(3,18,11,32,7,22,15,28)(4,31,16,17,8,27,12,21) );
G=PermutationGroup([(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,13,5,9),(2,10,6,14),(3,15,7,11),(4,12,8,16),(17,31,21,27),(18,28,22,32),(19,25,23,29),(20,30,24,26)], [(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)], [(1,20,9,26,5,24,13,30),(2,25,14,19,6,29,10,23),(3,18,11,32,7,22,15,28),(4,31,16,17,8,27,12,21)])
C2×C4.10D4 is a maximal subgroup of
(C2×Q8).Q8 C4.C22≀C2 (C23×C4).C4 2- 1+4⋊2C4 C42.97D4 (C2×Q8).211D4 C4.10D4⋊2C4 C4⋊Q8⋊15C4 C4.4D4⋊13C4 M4(2).45D4 M4(2).46D4 M4(2).49D4 C42.7D4 M4(2).50D4 C4.10D4⋊3C4 C42.114D4 C42.115D4 M4(2).31D4 M4(2).33D4 C42.129D4 C42.130D4 M4(2).5D4 M4(2).6D4 M4(2).7D4 C4⋊C4.97D4 C4⋊C4.98D4 M4(2).9D4 M4(2).11D4 M4(2).13D4 M4(2).15D4 (C2×C8).6D4 C4⋊Q8.C4 M4(2).25C23 M4(2).C23 M4(2).38D4 (C2×Q8).7F5
C2×C4.10D4 is a maximal quotient of
C42.394D4 (C2×C4)⋊M4(2) C42.44D4 C42.396D4 C24.45(C2×C4) C42.406D4 C42.408D4 C42.68D4 C42.71D4 C42.74D4 C42.412D4 C42.414D4 C42.416D4 C42.81D4 C42.83D4 C42.88D4 C4.C22≀C2 C42.97D4 (C22×C4).275D4 M4(2).45D4 C42.114D4 M4(2)⋊8Q8 (C2×Q8).7F5
Matrix representation of C2×C4.10D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 16 | 1 | 0 |
| 0 | 0 | 16 | 0 | 1 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 3 | 15 |
| 0 | 0 | 7 | 0 | 11 | 9 |
| 0 | 0 | 7 | 10 | 10 | 16 |
| 0 | 0 | 8 | 16 | 10 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,16,0,0,0,2,1,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,16,0,0,0,2,1,1,1,0,0,0,16,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,8,7,7,8,0,0,0,0,10,16,0,0,3,11,10,10,0,0,15,9,16,16] >;
C2×C4.10D4 in GAP, Magma, Sage, TeX
C_2\times C_4._{10}D_4 % in TeX
G:=Group("C2xC4.10D4"); // GroupNames label
G:=SmallGroup(64,93);
// by ID
G=gap.SmallGroup(64,93);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,963,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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