direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4×C4○D4, C22.10C25, C24.603C23, C42.749C23, C23.104C24, D4○(C2×C42), Q8○(C2×C42), C42○2(C4×D4), C42○2(C2×D4), C42○2(C2×Q8), C42○2(C4×Q8), C42○(C4○D4), D4⋊7(C22×C4), C2.6(C24×C4), Q8⋊7(C22×C4), C42○(C22×D4), C42○(C22×Q8), C4.36(C23×C4), (C4×D4)⋊111C22, C4⋊C4.512C23, (C2×C4).693C24, (C22×C42)⋊20C2, (C2×C42)⋊88C22, (C4×Q8)⋊101C22, C22.1(C23×C4), (C2×D4).495C23, (C2×Q8).478C23, C42○2(C42⋊C2), C22⋊C4.124C23, (C23×C4).661C22, C23.154(C22×C4), C42⋊C2⋊103C22, (C22×C4).1292C23, (C22×D4).612C22, (C22×Q8).512C22, C4○2(C2×C4×D4), C4○2(C2×C4×Q8), C4○(C4×C4○D4), C42○(C2×C4×D4), (C2×C4)○3(C4×D4), C42○(C2×C4×Q8), (C2×C4×D4)⋊96C2, (C2×C4)○3(C4×Q8), (C2×C4×Q8)⋊64C2, C4⋊C4○2(C2×C42), C42○4(C2×C4⋊C4), C42○(C4×C4○D4), (C2×C42)○(C4×D4), (C2×C42)○(C4×Q8), (C2×Q8)○(C2×C42), (C2×D4)⋊54(C2×C4), (C2×Q8)⋊45(C2×C4), C42○2(C2×C4○D4), (C2×D4)○2(C2×C42), C4○2(C2×C42⋊C2), (C22×C4)⋊48(C2×C4), (C2×C4)⋊10(C22×C4), C22⋊C4○2(C2×C42), C42○3(C2×C22⋊C4), C2.4(C22×C4○D4), C42○(C22×C4○D4), (C2×C42)○(C22×Q8), (C2×C4)○3(C42⋊C2), (C2×C42⋊C2)⋊69C2, (C2×C4⋊C4).981C22, C42○2(C2×C42⋊C2), (C22×C4○D4).30C2, C22.145(C2×C4○D4), (C2×C42)○(C42⋊C2), (C2×C4○D4).340C22, (C2×C22⋊C4).560C22, (C2×C4)○2(C2×C4×Q8), (C2×C4)○(C4×C4○D4), (C2×C42)○(C2×C4×Q8), (C2×C42)○(C2×C4⋊C4), (C2×C42)○(C2×C4○D4), (C2×C4)○2(C2×C42⋊C2), (C2×C42)○(C22×C4○D4), (C2×C42)○(C2×C42⋊C2), SmallGroup(128,2156)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4×C4○D4
G = < a,b,c,d,e | a2=b4=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: 1020 in 864 conjugacy classes, 708 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C22×C42, C2×C42⋊C2, C2×C4×D4, C2×C4×Q8, C4×C4○D4, C22×C4○D4, C2×C4×C4○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, C25, C4×C4○D4, C24×C4, C22×C4○D4, C2×C4×C4○D4
►On 64 points
(1 25)(2 26)(3 27)(4 28)(5 37)(6 38)(7 39)(8 40)(9 15)(10 16)(11 13)(12 14)(17 31)(18 32)(19 29)(20 30)(21 35)(22 36)(23 33)(24 34)(41 47)(42 48)(43 45)(44 46)(49 62)(50 63)(51 64)(52 61)(53 59)(54 60)(55 57)(56 58)
(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)
(1 29 11 7)(2 30 12 8)(3 31 9 5)(4 32 10 6)(13 39 25 19)(14 40 26 20)(15 37 27 17)(16 38 28 18)(21 57 52 45)(22 58 49 46)(23 59 50 47)(24 60 51 48)(33 53 63 41)(34 54 64 42)(35 55 61 43)(36 56 62 44)
(1 19 11 39)(2 20 12 40)(3 17 9 37)(4 18 10 38)(5 27 31 15)(6 28 32 16)(7 25 29 13)(8 26 30 14)(21 43 52 55)(22 44 49 56)(23 41 50 53)(24 42 51 54)(33 47 63 59)(34 48 64 60)(35 45 61 57)(36 46 62 58)
(1 35)(2 36)(3 33)(4 34)(5 41)(6 42)(7 43)(8 44)(9 63)(10 64)(11 61)(12 62)(13 52)(14 49)(15 50)(16 51)(17 59)(18 60)(19 57)(20 58)(21 25)(22 26)(23 27)(24 28)(29 55)(30 56)(31 53)(32 54)(37 47)(38 48)(39 45)(40 46)
G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,37)(6,38)(7,39)(8,40)(9,15)(10,16)(11,13)(12,14)(17,31)(18,32)(19,29)(20,30)(21,35)(22,36)(23,33)(24,34)(41,47)(42,48)(43,45)(44,46)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58), (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), (1,29,11,7)(2,30,12,8)(3,31,9,5)(4,32,10,6)(13,39,25,19)(14,40,26,20)(15,37,27,17)(16,38,28,18)(21,57,52,45)(22,58,49,46)(23,59,50,47)(24,60,51,48)(33,53,63,41)(34,54,64,42)(35,55,61,43)(36,56,62,44), (1,19,11,39)(2,20,12,40)(3,17,9,37)(4,18,10,38)(5,27,31,15)(6,28,32,16)(7,25,29,13)(8,26,30,14)(21,43,52,55)(22,44,49,56)(23,41,50,53)(24,42,51,54)(33,47,63,59)(34,48,64,60)(35,45,61,57)(36,46,62,58), (1,35)(2,36)(3,33)(4,34)(5,41)(6,42)(7,43)(8,44)(9,63)(10,64)(11,61)(12,62)(13,52)(14,49)(15,50)(16,51)(17,59)(18,60)(19,57)(20,58)(21,25)(22,26)(23,27)(24,28)(29,55)(30,56)(31,53)(32,54)(37,47)(38,48)(39,45)(40,46)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,37)(6,38)(7,39)(8,40)(9,15)(10,16)(11,13)(12,14)(17,31)(18,32)(19,29)(20,30)(21,35)(22,36)(23,33)(24,34)(41,47)(42,48)(43,45)(44,46)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58), (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), (1,29,11,7)(2,30,12,8)(3,31,9,5)(4,32,10,6)(13,39,25,19)(14,40,26,20)(15,37,27,17)(16,38,28,18)(21,57,52,45)(22,58,49,46)(23,59,50,47)(24,60,51,48)(33,53,63,41)(34,54,64,42)(35,55,61,43)(36,56,62,44), (1,19,11,39)(2,20,12,40)(3,17,9,37)(4,18,10,38)(5,27,31,15)(6,28,32,16)(7,25,29,13)(8,26,30,14)(21,43,52,55)(22,44,49,56)(23,41,50,53)(24,42,51,54)(33,47,63,59)(34,48,64,60)(35,45,61,57)(36,46,62,58), (1,35)(2,36)(3,33)(4,34)(5,41)(6,42)(7,43)(8,44)(9,63)(10,64)(11,61)(12,62)(13,52)(14,49)(15,50)(16,51)(17,59)(18,60)(19,57)(20,58)(21,25)(22,26)(23,27)(24,28)(29,55)(30,56)(31,53)(32,54)(37,47)(38,48)(39,45)(40,46) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,37),(6,38),(7,39),(8,40),(9,15),(10,16),(11,13),(12,14),(17,31),(18,32),(19,29),(20,30),(21,35),(22,36),(23,33),(24,34),(41,47),(42,48),(43,45),(44,46),(49,62),(50,63),(51,64),(52,61),(53,59),(54,60),(55,57),(56,58)], [(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)], [(1,29,11,7),(2,30,12,8),(3,31,9,5),(4,32,10,6),(13,39,25,19),(14,40,26,20),(15,37,27,17),(16,38,28,18),(21,57,52,45),(22,58,49,46),(23,59,50,47),(24,60,51,48),(33,53,63,41),(34,54,64,42),(35,55,61,43),(36,56,62,44)], [(1,19,11,39),(2,20,12,40),(3,17,9,37),(4,18,10,38),(5,27,31,15),(6,28,32,16),(7,25,29,13),(8,26,30,14),(21,43,52,55),(22,44,49,56),(23,41,50,53),(24,42,51,54),(33,47,63,59),(34,48,64,60),(35,45,61,57),(36,46,62,58)], [(1,35),(2,36),(3,33),(4,34),(5,41),(6,42),(7,43),(8,44),(9,63),(10,64),(11,61),(12,62),(13,52),(14,49),(15,50),(16,51),(17,59),(18,60),(19,57),(20,58),(21,25),(22,26),(23,27),(24,28),(29,55),(30,56),(31,53),(32,54),(37,47),(38,48),(39,45),(40,46)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4X | 4Y | ··· | 4BH |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 |
| kernel | C2×C4×C4○D4 | C22×C42 | C2×C42⋊C2 | C2×C4×D4 | C2×C4×Q8 | C4×C4○D4 | C22×C4○D4 | C2×C4○D4 | C2×C4 |
| # reps | 1 | 3 | 3 | 6 | 2 | 16 | 1 | 32 | 16 |
Matrix representation of C2×C4×C4○D4 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,1,0,0,0,0,3,0,0,0,0,2],[1,0,0,0,0,4,0,0,0,0,0,3,0,0,2,0] >;
C2×C4×C4○D4 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_4\circ D_4
% in TeX
G:=Group("C2xC4xC4oD4"); // GroupNames label
G:=SmallGroup(128,2156);
// by ID
G=gap.SmallGroup(128,2156);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,352,136]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=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