direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C4○D4, D4⋊3C23, C2.3C25, Q8⋊3C23, C4.18C24, C22.1C24, C24.33C22, C23.49C23, C4○(C22×D4), D4○(C22×C4), Q8○(C22×C4), C4○(C22×Q8), (C23×C4)⋊8C2, (C2×C4)⋊5C23, (C22×D4)⋊13C2, (C2×D4)⋊19C22, (C2×Q8)⋊19C22, (C22×Q8)⋊11C2, (C22×C4)⋊20C22, C4○(C2×C4○D4), (C2×C4)○2(C2×D4), (C2×C4)○2(C2×Q8), (C2×C4)○(C4○D4), (C22×C4)○(C2×Q8), (C2×C4)○(C22×Q8), (C22×C4)○(C22×Q8), (C2×C4)○(C2×C4○D4), SmallGroup(64,263)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×C4○D4
G = < a,b,c,d,e | a2=b2=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >
Subgroups: 505 in 445 conjugacy classes, 385 normal (6 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C23, C23, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C22×C4○D4
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, C25, C22×C4○D4
►On 32 points
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)
(1 23)(2 24)(3 21)(4 22)(5 12)(6 9)(7 10)(8 11)(13 26)(14 27)(15 28)(16 25)(17 32)(18 29)(19 30)(20 31)
(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 29 3 31)(2 30 4 32)(5 16 7 14)(6 13 8 15)(9 26 11 28)(10 27 12 25)(17 24 19 22)(18 21 20 23)
(1 20)(2 17)(3 18)(4 19)(5 27)(6 28)(7 25)(8 26)(9 15)(10 16)(11 13)(12 14)(21 29)(22 30)(23 31)(24 32)
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,23)(2,24)(3,21)(4,22)(5,12)(6,9)(7,10)(8,11)(13,26)(14,27)(15,28)(16,25)(17,32)(18,29)(19,30)(20,31), (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), (1,29,3,31)(2,30,4,32)(5,16,7,14)(6,13,8,15)(9,26,11,28)(10,27,12,25)(17,24,19,22)(18,21,20,23), (1,20)(2,17)(3,18)(4,19)(5,27)(6,28)(7,25)(8,26)(9,15)(10,16)(11,13)(12,14)(21,29)(22,30)(23,31)(24,32)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,23)(2,24)(3,21)(4,22)(5,12)(6,9)(7,10)(8,11)(13,26)(14,27)(15,28)(16,25)(17,32)(18,29)(19,30)(20,31), (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), (1,29,3,31)(2,30,4,32)(5,16,7,14)(6,13,8,15)(9,26,11,28)(10,27,12,25)(17,24,19,22)(18,21,20,23), (1,20)(2,17)(3,18)(4,19)(5,27)(6,28)(7,25)(8,26)(9,15)(10,16)(11,13)(12,14)(21,29)(22,30)(23,31)(24,32) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,18),(14,19),(15,20),(16,17),(25,32),(26,29),(27,30),(28,31)], [(1,23),(2,24),(3,21),(4,22),(5,12),(6,9),(7,10),(8,11),(13,26),(14,27),(15,28),(16,25),(17,32),(18,29),(19,30),(20,31)], [(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)], [(1,29,3,31),(2,30,4,32),(5,16,7,14),(6,13,8,15),(9,26,11,28),(10,27,12,25),(17,24,19,22),(18,21,20,23)], [(1,20),(2,17),(3,18),(4,19),(5,27),(6,28),(7,25),(8,26),(9,15),(10,16),(11,13),(12,14),(21,29),(22,30),(23,31),(24,32)]])
C22×C4○D4 is a maximal subgroup of
C24.51(C2×C4) C24.165C23 (C23×C4).C4 C24.65D4 C24.66D4 C23.179C24 C24.542C23 C24.549C23 C23.223C24 C23.288C24 C23.304C24 C24.243C23 C24.244C23 C24.262C23 C24.263C23 C24.264C23 C24.360C23 C24.361C23 C24.73(C2×C4) D4○(C22⋊C8) C24.98D4 C24.103D4 C24.104D4 C24.105D4 C24.106D4 C22.14C25 C22.38C25 C22.74C25 C22.76C25 C22.77C25 C22.78C25 C4○D4⋊A4
C22×C4○D4 is a maximal quotient of
D4×C22×C4 Q8×C22×C4 C22.33C25 C22.44C25 C22.48C25 C22.49C25 C22.50C25 C22.64C25 C22.69C25 C22.70C25 C22.71C25 C22.72C25 C22.79C25 C22.80C25 C22.81C25 C22.82C25 C22.83C25 C22.84C25 C22.94C25 C22.95C25 C22.96C25 C22.97C25 C22.98C25 C22.99C25 C22.100C25 C22.101C25 C22.102C25 C22.103C25 C22.104C25 C22.105C25 C22.106C25 C22.107C25 C22.108C25 C23.144C24 C22.110C25 C22.111C25 C23.146C24 C22.113C25
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4H | 4I | ··· | 4T |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C4○D4 |
| kernel | C22×C4○D4 | C23×C4 | C22×D4 | C22×Q8 | C2×C4○D4 | C22 |
| # reps | 1 | 3 | 3 | 1 | 24 | 8 |
Matrix representation of C22×C4○D4 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,4,0] >;
C22×C4○D4 in GAP, Magma, Sage, TeX
C_2^2\times C_4\circ D_4
% in TeX
G:=Group("C2^2xC4oD4"); // GroupNames label
G:=SmallGroup(64,263);
// by ID
G=gap.SmallGroup(64,263);
# by ID
G:=PCGroup([6,-2,2,2,2,2,-2,409,158]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations