p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.440D4, C24.600C23, C23.757C24, C22⋊1(C4⋊Q8), C4⋊4(C22⋊Q8), (C22×C4)⋊22Q8, C42⋊9C4⋊41C2, (C22×C4).599D4, C23.375(C2×D4), C23.106(C2×Q8), C4.103(C4⋊D4), (C22×C42).28C2, (C22×C4).264C23, (C23×C4).684C22, C22.467(C22×D4), C23.7Q8.80C2, C22.181(C22×Q8), (C2×C42).1092C22, (C22×Q8).249C22, C23.67C23⋊110C2, C23.65C23⋊170C2, C2.C42.453C22, C2.57(C22.26C24), C2.47(C23.37C23), (C2×C4⋊Q8)⋊26C2, C2.21(C2×C4⋊Q8), C2.50(C2×C4⋊D4), (C2×C4).173(C2×Q8), C2.48(C2×C22⋊Q8), (C2×C4).1396(C2×D4), (C2×C22⋊Q8).50C2, (C2×C4).673(C4○D4), (C2×C4⋊C4).560C22, C22.598(C2×C4○D4), (C2×C22⋊C4).367C22, SmallGroup(128,1589)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.440D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 532 in 322 conjugacy classes, 144 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C4⋊Q8, C23×C4, C23×C4, C22×Q8, C23.7Q8, C42⋊9C4, C23.65C23, C23.67C23, C22×C42, C2×C22⋊Q8, C2×C4⋊Q8, C42.440D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C4⋊D4, C22⋊Q8, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C4⋊D4, C2×C22⋊Q8, C2×C4⋊Q8, C22.26C24, C23.37C23, C42.440D4
►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 13 9 25)(2 14 10 26)(3 15 11 27)(4 16 12 28)(5 19 23 64)(6 20 24 61)(7 17 21 62)(8 18 22 63)(29 35 39 49)(30 36 40 50)(31 33 37 51)(32 34 38 52)(41 53 57 45)(42 54 58 46)(43 55 59 47)(44 56 60 48)
(1 20 31 55)(2 19 32 54)(3 18 29 53)(4 17 30 56)(5 34 42 14)(6 33 43 13)(7 36 44 16)(8 35 41 15)(9 61 37 47)(10 64 38 46)(11 63 39 45)(12 62 40 48)(21 50 60 28)(22 49 57 27)(23 52 58 26)(24 51 59 25)
(1 45 3 47)(2 48 4 46)(5 52 7 50)(6 51 8 49)(9 53 11 55)(10 56 12 54)(13 57 15 59)(14 60 16 58)(17 40 19 38)(18 39 20 37)(21 36 23 34)(22 35 24 33)(25 41 27 43)(26 44 28 42)(29 61 31 63)(30 64 32 62)
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), (1,13,9,25)(2,14,10,26)(3,15,11,27)(4,16,12,28)(5,19,23,64)(6,20,24,61)(7,17,21,62)(8,18,22,63)(29,35,39,49)(30,36,40,50)(31,33,37,51)(32,34,38,52)(41,53,57,45)(42,54,58,46)(43,55,59,47)(44,56,60,48), (1,20,31,55)(2,19,32,54)(3,18,29,53)(4,17,30,56)(5,34,42,14)(6,33,43,13)(7,36,44,16)(8,35,41,15)(9,61,37,47)(10,64,38,46)(11,63,39,45)(12,62,40,48)(21,50,60,28)(22,49,57,27)(23,52,58,26)(24,51,59,25), (1,45,3,47)(2,48,4,46)(5,52,7,50)(6,51,8,49)(9,53,11,55)(10,56,12,54)(13,57,15,59)(14,60,16,58)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,41,27,43)(26,44,28,42)(29,61,31,63)(30,64,32,62)>;
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), (1,13,9,25)(2,14,10,26)(3,15,11,27)(4,16,12,28)(5,19,23,64)(6,20,24,61)(7,17,21,62)(8,18,22,63)(29,35,39,49)(30,36,40,50)(31,33,37,51)(32,34,38,52)(41,53,57,45)(42,54,58,46)(43,55,59,47)(44,56,60,48), (1,20,31,55)(2,19,32,54)(3,18,29,53)(4,17,30,56)(5,34,42,14)(6,33,43,13)(7,36,44,16)(8,35,41,15)(9,61,37,47)(10,64,38,46)(11,63,39,45)(12,62,40,48)(21,50,60,28)(22,49,57,27)(23,52,58,26)(24,51,59,25), (1,45,3,47)(2,48,4,46)(5,52,7,50)(6,51,8,49)(9,53,11,55)(10,56,12,54)(13,57,15,59)(14,60,16,58)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,41,27,43)(26,44,28,42)(29,61,31,63)(30,64,32,62) );
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)], [(1,13,9,25),(2,14,10,26),(3,15,11,27),(4,16,12,28),(5,19,23,64),(6,20,24,61),(7,17,21,62),(8,18,22,63),(29,35,39,49),(30,36,40,50),(31,33,37,51),(32,34,38,52),(41,53,57,45),(42,54,58,46),(43,55,59,47),(44,56,60,48)], [(1,20,31,55),(2,19,32,54),(3,18,29,53),(4,17,30,56),(5,34,42,14),(6,33,43,13),(7,36,44,16),(8,35,41,15),(9,61,37,47),(10,64,38,46),(11,63,39,45),(12,62,40,48),(21,50,60,28),(22,49,57,27),(23,52,58,26),(24,51,59,25)], [(1,45,3,47),(2,48,4,46),(5,52,7,50),(6,51,8,49),(9,53,11,55),(10,56,12,54),(13,57,15,59),(14,60,16,58),(17,40,19,38),(18,39,20,37),(21,36,23,34),(22,35,24,33),(25,41,27,43),(26,44,28,42),(29,61,31,63),(30,64,32,62)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4X | 4Y | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 |
| kernel | C42.440D4 | C23.7Q8 | C42⋊9C4 | C23.65C23 | C23.67C23 | C22×C42 | C2×C22⋊Q8 | C2×C4⋊Q8 | C42 | C22×C4 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 4 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 12 |
Matrix representation of C42.440D4 ►in GL6(𝔽5)
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C42.440D4 in GAP, Magma, Sage, TeX
C_4^2._{440}D_4 % in TeX
G:=Group("C4^2.440D4"); // GroupNames label
G:=SmallGroup(128,1589);
// by ID
G=gap.SmallGroup(128,1589);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,184,2019]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations