direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×Q8⋊6D4, C23.24C24, C22.56C25, C42.555C23, C24.493C23, C22.1102+ (1+4), (C2×Q8)⋊43D4, Q8⋊11(C2×D4), (C4×Q8)⋊92C22, C2.23(D4×C23), (C4×D4)⋊104C22, C4⋊D4⋊72C22, C4⋊1D4⋊47C22, C4⋊C4.468C23, (C2×C4).598C24, C4.112(C22×D4), (C2×D4).452C23, C22⋊C4.83C23, (C2×Q8).485C23, (C2×C42).927C22, (C23×C4).596C22, C22.163(C22×D4), C2.17(C2×2+ (1+4)), (C22×C4).1193C23, (C22×D4).422C22, (C22×Q8).514C22, C4⋊C4○3(C2×Q8), Q8○2(C2×C4⋊C4), (C2×C4×D4)⋊82C2, C4⋊3(C2×C4○D4), (C2×C4×Q8)⋊51C2, C4⋊C4○(C22×Q8), (C2×C4)⋊20(C4○D4), (C2×C4⋊1D4)⋊25C2, (C2×C4⋊D4)⋊62C2, (C2×C4).1112(C2×D4), (C2×C4○D4)⋊73C22, (C22×C4○D4)⋊19C2, C2.28(C22×C4○D4), (C2×C4⋊C4).956C22, C22.158(C2×C4○D4), (C2×C22⋊C4).539C22, (C2×Q8)○2(C2×C4⋊C4), (C2×C4⋊C4)○(C22×Q8), SmallGroup(128,2199)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 1436 in 868 conjugacy classes, 444 normal (11 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×16], C4 [×10], C22, C22 [×6], C22 [×60], C2×C4 [×34], C2×C4 [×58], D4 [×96], Q8 [×16], C23, C23 [×12], C23 [×36], C42 [×12], C22⋊C4 [×24], C4⋊C4 [×16], C22×C4, C22×C4 [×32], C22×C4 [×24], C2×D4 [×60], C2×D4 [×48], C2×Q8 [×12], C4○D4 [×64], C24 [×6], C2×C42 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×3], C4×D4 [×24], C4×Q8 [×8], C4⋊D4 [×48], C4⋊1D4 [×24], C23×C4 [×6], C22×D4 [×15], C22×Q8, C2×C4○D4 [×16], C2×C4○D4 [×16], C2×C4×D4 [×3], C2×C4×Q8, C2×C4⋊D4 [×6], C2×C4⋊1D4 [×3], Q8⋊6D4 [×16], C22×C4○D4 [×2], C2×Q8⋊6D4
Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×4], C24 [×31], C22×D4 [×14], C2×C4○D4 [×6], 2+ (1+4) [×2], C25, Q8⋊6D4 [×4], D4×C23, C22×C4○D4, C2×2+ (1+4), C2×Q8⋊6D4
Generators and relations
G = < a,b,c,d,e | a2=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
►On 64 points
(1 7)(2 8)(3 5)(4 6)(9 22)(10 23)(11 24)(12 21)(13 49)(14 50)(15 51)(16 52)(17 40)(18 37)(19 38)(20 39)(25 30)(26 31)(27 32)(28 29)(33 54)(34 55)(35 56)(36 53)(41 48)(42 45)(43 46)(44 47)(57 62)(58 63)(59 64)(60 61)
(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 17 3 19)(2 20 4 18)(5 38 7 40)(6 37 8 39)(9 26 11 28)(10 25 12 27)(13 35 15 33)(14 34 16 36)(21 32 23 30)(22 31 24 29)(41 62 43 64)(42 61 44 63)(45 60 47 58)(46 59 48 57)(49 56 51 54)(50 55 52 53)
(1 52 24 64)(2 51 21 63)(3 50 22 62)(4 49 23 61)(5 14 9 57)(6 13 10 60)(7 16 11 59)(8 15 12 58)(17 55 29 43)(18 54 30 42)(19 53 31 41)(20 56 32 44)(25 45 37 33)(26 48 38 36)(27 47 39 35)(28 46 40 34)
(1 47)(2 48)(3 45)(4 46)(5 42)(6 43)(7 44)(8 41)(9 54)(10 55)(11 56)(12 53)(13 29)(14 30)(15 31)(16 32)(17 60)(18 57)(19 58)(20 59)(21 36)(22 33)(23 34)(24 35)(25 50)(26 51)(27 52)(28 49)(37 62)(38 63)(39 64)(40 61)
G:=sub<Sym(64)| (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,21)(13,49)(14,50)(15,51)(16,52)(17,40)(18,37)(19,38)(20,39)(25,30)(26,31)(27,32)(28,29)(33,54)(34,55)(35,56)(36,53)(41,48)(42,45)(43,46)(44,47)(57,62)(58,63)(59,64)(60,61), (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), (1,17,3,19)(2,20,4,18)(5,38,7,40)(6,37,8,39)(9,26,11,28)(10,25,12,27)(13,35,15,33)(14,34,16,36)(21,32,23,30)(22,31,24,29)(41,62,43,64)(42,61,44,63)(45,60,47,58)(46,59,48,57)(49,56,51,54)(50,55,52,53), (1,52,24,64)(2,51,21,63)(3,50,22,62)(4,49,23,61)(5,14,9,57)(6,13,10,60)(7,16,11,59)(8,15,12,58)(17,55,29,43)(18,54,30,42)(19,53,31,41)(20,56,32,44)(25,45,37,33)(26,48,38,36)(27,47,39,35)(28,46,40,34), (1,47)(2,48)(3,45)(4,46)(5,42)(6,43)(7,44)(8,41)(9,54)(10,55)(11,56)(12,53)(13,29)(14,30)(15,31)(16,32)(17,60)(18,57)(19,58)(20,59)(21,36)(22,33)(23,34)(24,35)(25,50)(26,51)(27,52)(28,49)(37,62)(38,63)(39,64)(40,61)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,21)(13,49)(14,50)(15,51)(16,52)(17,40)(18,37)(19,38)(20,39)(25,30)(26,31)(27,32)(28,29)(33,54)(34,55)(35,56)(36,53)(41,48)(42,45)(43,46)(44,47)(57,62)(58,63)(59,64)(60,61), (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), (1,17,3,19)(2,20,4,18)(5,38,7,40)(6,37,8,39)(9,26,11,28)(10,25,12,27)(13,35,15,33)(14,34,16,36)(21,32,23,30)(22,31,24,29)(41,62,43,64)(42,61,44,63)(45,60,47,58)(46,59,48,57)(49,56,51,54)(50,55,52,53), (1,52,24,64)(2,51,21,63)(3,50,22,62)(4,49,23,61)(5,14,9,57)(6,13,10,60)(7,16,11,59)(8,15,12,58)(17,55,29,43)(18,54,30,42)(19,53,31,41)(20,56,32,44)(25,45,37,33)(26,48,38,36)(27,47,39,35)(28,46,40,34), (1,47)(2,48)(3,45)(4,46)(5,42)(6,43)(7,44)(8,41)(9,54)(10,55)(11,56)(12,53)(13,29)(14,30)(15,31)(16,32)(17,60)(18,57)(19,58)(20,59)(21,36)(22,33)(23,34)(24,35)(25,50)(26,51)(27,52)(28,49)(37,62)(38,63)(39,64)(40,61) );
G=PermutationGroup([(1,7),(2,8),(3,5),(4,6),(9,22),(10,23),(11,24),(12,21),(13,49),(14,50),(15,51),(16,52),(17,40),(18,37),(19,38),(20,39),(25,30),(26,31),(27,32),(28,29),(33,54),(34,55),(35,56),(36,53),(41,48),(42,45),(43,46),(44,47),(57,62),(58,63),(59,64),(60,61)], [(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)], [(1,17,3,19),(2,20,4,18),(5,38,7,40),(6,37,8,39),(9,26,11,28),(10,25,12,27),(13,35,15,33),(14,34,16,36),(21,32,23,30),(22,31,24,29),(41,62,43,64),(42,61,44,63),(45,60,47,58),(46,59,48,57),(49,56,51,54),(50,55,52,53)], [(1,52,24,64),(2,51,21,63),(3,50,22,62),(4,49,23,61),(5,14,9,57),(6,13,10,60),(7,16,11,59),(8,15,12,58),(17,55,29,43),(18,54,30,42),(19,53,31,41),(20,56,32,44),(25,45,37,33),(26,48,38,36),(27,47,39,35),(28,46,40,34)], [(1,47),(2,48),(3,45),(4,46),(5,42),(6,43),(7,44),(8,41),(9,54),(10,55),(11,56),(12,53),(13,29),(14,30),(15,31),(16,32),(17,60),(18,57),(19,58),(20,59),(21,36),(22,33),(23,34),(24,35),(25,50),(26,51),(27,52),(28,49),(37,62),(38,63),(39,64),(40,61)])
Matrix representation ►G ⊆ GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 1 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,4,3],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,3,2,0,0,0,0,2],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,3,1],[4,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,4] >;
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4X | 4Y | ··· | 4AD |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ (1+4) |
| kernel | C2×Q8⋊6D4 | C2×C4×D4 | C2×C4×Q8 | C2×C4⋊D4 | C2×C4⋊1D4 | Q8⋊6D4 | C22×C4○D4 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 3 | 1 | 6 | 3 | 16 | 2 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_6D_4
% in TeX
G:=Group("C2xQ8:6D4"); // GroupNames label
G:=SmallGroup(128,2199);
// by ID
G=gap.SmallGroup(128,2199);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,184,570,136]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations