p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.12C4, C4.12M4(2), C42.92C22, (C2×C4)⋊4C8, (C4×C8)⋊2C2, C4○2(C4⋊C8), C4⋊C8⋊17C2, C4.9(C2×C8), C42○(C4⋊C8), C4○2(C22⋊C8), C22⋊C8.9C2, C2.3(C22×C8), C22.5(C2×C8), C4.48(C4○D4), C42○(C22⋊C8), (C22×C4).16C4, (C2×C8).61C22, (C2×C42).15C2, C23.32(C2×C4), C2.5(C2×M4(2)), (C2×C4).150C23, C2.4(C42⋊C2), C22.22(C22×C4), (C22×C4).93C22, (C2×C4).57(C2×C4), SmallGroup(64,112)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.12C4
G = < a,b,c | a4=b4=1, c4=a2, ab=ba, cac-1=a-1b2, bc=cb >
Subgroups: 73 in 59 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C42.12C4
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, C42⋊C2, C22×C8, C2×M4(2), C42.12C4
►On 32 points
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 11 13 15)(10 24 14 20)(12 18 16 22)(17 19 21 23)(25 27 29 31)
(1 21 27 13)(2 22 28 14)(3 23 29 15)(4 24 30 16)(5 17 31 9)(6 18 32 10)(7 19 25 11)(8 20 26 12)
(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)
G:=sub<Sym(32)| (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,11,13,15)(10,24,14,20)(12,18,16,22)(17,19,21,23)(25,27,29,31), (1,21,27,13)(2,22,28,14)(3,23,29,15)(4,24,30,16)(5,17,31,9)(6,18,32,10)(7,19,25,11)(8,20,26,12), (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)>;
G:=Group( (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,11,13,15)(10,24,14,20)(12,18,16,22)(17,19,21,23)(25,27,29,31), (1,21,27,13)(2,22,28,14)(3,23,29,15)(4,24,30,16)(5,17,31,9)(6,18,32,10)(7,19,25,11)(8,20,26,12), (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) );
G=PermutationGroup([[(1,3,5,7),(2,26,6,30),(4,28,8,32),(9,11,13,15),(10,24,14,20),(12,18,16,22),(17,19,21,23),(25,27,29,31)], [(1,21,27,13),(2,22,28,14),(3,23,29,15),(4,24,30,16),(5,17,31,9),(6,18,32,10),(7,19,25,11),(8,20,26,12)], [(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)]])
C42.12C4 is a maximal subgroup of
C8⋊9M4(2) C23.27C42 C82⋊15C2 C82⋊2C2 C42.677C23 C42.260C23 C42.262C23 C42.678C23 C42.290C23 C42.293C23 C42.294C23 D4⋊7M4(2) C42.693C23 C42.694C23 C42.301C23 C42.302C23 Q8.4M4(2) C42.696C23 C42.304C23 C42.698C23 D4⋊8M4(2) Q8⋊7M4(2) C42.308C23 C42.309C23
C42.D2p: C42⋊1C8 C42.20D4 C42⋊6C8 M4(2)⋊C8 C42.2Q8 C42.3Q8 C42.5Q8 C42.23D4 ...
C2p.(C22×C8): C8×M4(2) C82⋊C2 C8×C4○D4 C42.691C23 C42.695C23 C42.697C23 Dic3.5M4(2) Dic5.14M4(2) ...
C42.12C4 is a maximal quotient of
C42⋊4C8 C23.32M4(2) C42⋊5C8 C42.13C8 C42.6C8 C8.12M4(2) C42.6F5 C42.12F5 Dic5.12M4(2) C20.34M4(2)
C42.D2p: C4×C22⋊C8 C4×C4⋊C8 C42.425D4 C42⋊8C8 C42.282D6 C42.200D6 C42.285D6 C42.282D10 ...
(C2×C8).D2p: C4⋊C4⋊3C8 C22⋊C4⋊4C8 Dic3.5M4(2) Dic5.14M4(2) Dic7.5M4(2) ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | M4(2) | C4○D4 |
| kernel | C42.12C4 | C4×C8 | C22⋊C8 | C4⋊C8 | C2×C42 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 16 | 4 | 4 |
Matrix representation of C42.12C4 ►in GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 4 | 1 |
| 13 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 15 | 0 | 0 |
| 0 | 13 | 15 |
| 0 | 0 | 4 |
G:=sub<GL(3,GF(17))| [13,0,0,0,16,4,0,0,1],[13,0,0,0,13,0,0,0,13],[15,0,0,0,13,0,0,15,4] >;
C42.12C4 in GAP, Magma, Sage, TeX
C_4^2._{12}C_4 % in TeX
G:=Group("C4^2.12C4"); // GroupNames label
G:=SmallGroup(64,112);
// by ID
G=gap.SmallGroup(64,112);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,50,88]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,b*c=c*b>;
// generators/relations