p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊15D4, C24.235C23, C23.292C24, (C22×C4)⋊18D4, C4○2(C23⋊2D4), C23.145(C2×D4), C4○(C23.Q8), C4.209(C4⋊D4), C4○(C23.11D4), C23.15(C4○D4), C23⋊2D4.39C2, C4○2(C23.10D4), (C23×C4).323C22, (C2×C42).455C22, (C22×C4).496C23, C23.Q8⋊102C2, C22.175(C22×D4), C23.10D4⋊117C2, C23.11D4⋊139C2, C4○(C23.81C23), (C22×D4).494C22, C2.10(C22.19C24), C2.9(C22.26C24), C23.81C23⋊145C2, C2.C42.532C22, C2.12(C23.36C23), (C2×C4×D4)⋊17C2, (C4×C4⋊C4)⋊51C2, (C4×C22⋊C4)⋊47C2, (C2×C4).293(C2×D4), C2.12(C2×C4⋊D4), (C2×C4).87(C4○D4), (C2×C42⋊C2)⋊16C2, (C2×C4)○2(C23⋊2D4), (C2×C4⋊C4).836C22, C22.172(C2×C4○D4), (C2×C4)○(C23.Q8), (C2×C4)○(C23.10D4), (C2×C22⋊C4).486C22, (C22×C4)○(C23.Q8), SmallGroup(128,1124)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊15D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, bd=db, dcd=c-1 >
Subgroups: 644 in 348 conjugacy classes, 116 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C23×C4, C23×C4, C22×D4, C22×D4, C4×C22⋊C4, C4×C4⋊C4, C23⋊2D4, C23.10D4, C23.10D4, C23.Q8, C23.11D4, C23.81C23, C2×C42⋊C2, C2×C4×D4, C2×C4×D4, C42⋊15D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, C22.19C24, C23.36C23, C22.26C24, C42⋊15D4
►On 64 points
(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 7 23 9)(2 8 24 10)(3 5 21 11)(4 6 22 12)(13 31 27 37)(14 32 28 38)(15 29 25 39)(16 30 26 40)(17 42 64 60)(18 43 61 57)(19 44 62 58)(20 41 63 59)(33 49 48 54)(34 50 45 55)(35 51 46 56)(36 52 47 53)
(1 58 25 35)(2 43 26 45)(3 60 27 33)(4 41 28 47)(5 17 37 49)(6 63 38 53)(7 19 39 51)(8 61 40 55)(9 62 29 56)(10 18 30 50)(11 64 31 54)(12 20 32 52)(13 48 21 42)(14 36 22 59)(15 46 23 44)(16 34 24 57)
(1 45)(2 35)(3 47)(4 33)(5 53)(6 49)(7 55)(8 51)(9 50)(10 56)(11 52)(12 54)(13 59)(14 42)(15 57)(16 44)(17 38)(18 29)(19 40)(20 31)(21 36)(22 48)(23 34)(24 46)(25 43)(26 58)(27 41)(28 60)(30 62)(32 64)(37 63)(39 61)
G:=sub<Sym(64)| (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), (1,7,23,9)(2,8,24,10)(3,5,21,11)(4,6,22,12)(13,31,27,37)(14,32,28,38)(15,29,25,39)(16,30,26,40)(17,42,64,60)(18,43,61,57)(19,44,62,58)(20,41,63,59)(33,49,48,54)(34,50,45,55)(35,51,46,56)(36,52,47,53), (1,58,25,35)(2,43,26,45)(3,60,27,33)(4,41,28,47)(5,17,37,49)(6,63,38,53)(7,19,39,51)(8,61,40,55)(9,62,29,56)(10,18,30,50)(11,64,31,54)(12,20,32,52)(13,48,21,42)(14,36,22,59)(15,46,23,44)(16,34,24,57), (1,45)(2,35)(3,47)(4,33)(5,53)(6,49)(7,55)(8,51)(9,50)(10,56)(11,52)(12,54)(13,59)(14,42)(15,57)(16,44)(17,38)(18,29)(19,40)(20,31)(21,36)(22,48)(23,34)(24,46)(25,43)(26,58)(27,41)(28,60)(30,62)(32,64)(37,63)(39,61)>;
G:=Group( (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), (1,7,23,9)(2,8,24,10)(3,5,21,11)(4,6,22,12)(13,31,27,37)(14,32,28,38)(15,29,25,39)(16,30,26,40)(17,42,64,60)(18,43,61,57)(19,44,62,58)(20,41,63,59)(33,49,48,54)(34,50,45,55)(35,51,46,56)(36,52,47,53), (1,58,25,35)(2,43,26,45)(3,60,27,33)(4,41,28,47)(5,17,37,49)(6,63,38,53)(7,19,39,51)(8,61,40,55)(9,62,29,56)(10,18,30,50)(11,64,31,54)(12,20,32,52)(13,48,21,42)(14,36,22,59)(15,46,23,44)(16,34,24,57), (1,45)(2,35)(3,47)(4,33)(5,53)(6,49)(7,55)(8,51)(9,50)(10,56)(11,52)(12,54)(13,59)(14,42)(15,57)(16,44)(17,38)(18,29)(19,40)(20,31)(21,36)(22,48)(23,34)(24,46)(25,43)(26,58)(27,41)(28,60)(30,62)(32,64)(37,63)(39,61) );
G=PermutationGroup([[(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)], [(1,7,23,9),(2,8,24,10),(3,5,21,11),(4,6,22,12),(13,31,27,37),(14,32,28,38),(15,29,25,39),(16,30,26,40),(17,42,64,60),(18,43,61,57),(19,44,62,58),(20,41,63,59),(33,49,48,54),(34,50,45,55),(35,51,46,56),(36,52,47,53)], [(1,58,25,35),(2,43,26,45),(3,60,27,33),(4,41,28,47),(5,17,37,49),(6,63,38,53),(7,19,39,51),(8,61,40,55),(9,62,29,56),(10,18,30,50),(11,64,31,54),(12,20,32,52),(13,48,21,42),(14,36,22,59),(15,46,23,44),(16,34,24,57)], [(1,45),(2,35),(3,47),(4,33),(5,53),(6,49),(7,55),(8,51),(9,50),(10,56),(11,52),(12,54),(13,59),(14,42),(15,57),(16,44),(17,38),(18,29),(19,40),(20,31),(21,36),(22,48),(23,34),(24,46),(25,43),(26,58),(27,41),(28,60),(30,62),(32,64),(37,63),(39,61)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 4A | ··· | 4H | 4I | ··· | 4AD |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 |
| kernel | C42⋊15D4 | C4×C22⋊C4 | C4×C4⋊C4 | C23⋊2D4 | C23.10D4 | C23.Q8 | C23.11D4 | C23.81C23 | C2×C42⋊C2 | C2×C4×D4 | C42 | C22×C4 | C2×C4 | C23 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 3 | 4 | 4 | 12 | 8 |
Matrix representation of C42⋊15D4 ►in GL6(𝔽5)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,3,1] >;
C42⋊15D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{15}D_4 % in TeX
G:=Group("C4^2:15D4"); // GroupNames label
G:=SmallGroup(128,1124);
// by ID
G=gap.SmallGroup(128,1124);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations