p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.358C23, (C4×D8)⋊6C2, (C4×Q16)⋊5C2, C4⋊C4.351D4, C4.Q16⋊25C2, D4⋊Q8⋊25C2, D4⋊D4.2C2, C2.20(D4○D8), C4⋊C8.57C22, C4⋊C4.77C23, (C4×C8).71C22, (C2×C8).51C23, C2.19(Q8○D8), D4.14(C4○D4), D4.2D4⋊21C2, D4.7D4⋊20C2, (C2×C4).322C24, Q8.12(C4○D4), Q8.D4⋊21C2, C22⋊C4.152D4, (C4×D4).84C22, C23.261(C2×D4), C4⋊Q8.107C22, SD16⋊C4⋊17C2, (C4×Q8).80C22, C8⋊C4.14C22, C2.D8.92C22, (C2×D8).128C22, (C2×D4).412C23, C22.D8⋊18C2, C4⋊D4.30C22, C22⋊C8.35C22, (C2×Q8).382C23, C22⋊Q8.30C22, C23.48D4⋊18C2, (C22×C4).295C23, C42.7C22⋊7C2, Q8⋊C4.38C22, (C2×Q16).123C22, (C2×SD16).19C22, C4.4D4.29C22, C22.582(C22×D4), D4⋊C4.164C22, C22.36C24⋊2C2, C42⋊C2.133C22, C23.33C23⋊13C2, C2.123(C22.19C24), C4.207(C2×C4○D4), (C2×C4).506(C2×D4), (C2×C4⋊C4).616C22, (C2×C4○D4).145C22, SmallGroup(128,1856)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 364 in 192 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×2], D4 [×8], Q8 [×2], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8, C4○D4 [×4], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2.D8 [×4], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×D4 [×2], C4×Q8 [×3], C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C2×D8, C2×SD16 [×2], C2×Q16, C2×C4○D4, C42.7C22, C4×D8, C4×Q16, SD16⋊C4 [×2], D4⋊D4, D4.7D4, D4.2D4, Q8.D4, D4⋊Q8, C4.Q16, C22.D8, C23.48D4, C23.33C23, C22.36C24, C42.358C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○D8, Q8○D8, C42.358C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2b2c, ece=bc, de=ed >
►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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 9 15 63)(6 10 16 64)(7 11 13 61)(8 12 14 62)(29 39 34 41)(30 40 35 42)(31 37 36 43)(32 38 33 44)(45 54 50 60)(46 55 51 57)(47 56 52 58)(48 53 49 59)
(5 11)(6 12)(7 9)(8 10)(13 63)(14 64)(15 61)(16 62)(21 26)(22 27)(23 28)(24 25)(29 39)(30 40)(31 37)(32 38)(33 44)(34 41)(35 42)(36 43)(45 52)(46 49)(47 50)(48 51)(53 55)(54 56)(57 59)(58 60)
(1 50 20 45)(2 46 17 51)(3 52 18 47)(4 48 19 49)(5 40 15 42)(6 43 16 37)(7 38 13 44)(8 41 14 39)(9 35 63 30)(10 31 64 36)(11 33 61 32)(12 29 62 34)(21 56 26 58)(22 59 27 53)(23 54 28 60)(24 57 25 55)
(1 32)(2 29)(3 30)(4 31)(5 58)(6 59)(7 60)(8 57)(9 52)(10 49)(11 50)(12 51)(13 54)(14 55)(15 56)(16 53)(17 34)(18 35)(19 36)(20 33)(21 40)(22 37)(23 38)(24 39)(25 41)(26 42)(27 43)(28 44)(45 61)(46 62)(47 63)(48 64)
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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,9,15,63)(6,10,16,64)(7,11,13,61)(8,12,14,62)(29,39,34,41)(30,40,35,42)(31,37,36,43)(32,38,33,44)(45,54,50,60)(46,55,51,57)(47,56,52,58)(48,53,49,59), (5,11)(6,12)(7,9)(8,10)(13,63)(14,64)(15,61)(16,62)(21,26)(22,27)(23,28)(24,25)(29,39)(30,40)(31,37)(32,38)(33,44)(34,41)(35,42)(36,43)(45,52)(46,49)(47,50)(48,51)(53,55)(54,56)(57,59)(58,60), (1,50,20,45)(2,46,17,51)(3,52,18,47)(4,48,19,49)(5,40,15,42)(6,43,16,37)(7,38,13,44)(8,41,14,39)(9,35,63,30)(10,31,64,36)(11,33,61,32)(12,29,62,34)(21,56,26,58)(22,59,27,53)(23,54,28,60)(24,57,25,55), (1,32)(2,29)(3,30)(4,31)(5,58)(6,59)(7,60)(8,57)(9,52)(10,49)(11,50)(12,51)(13,54)(14,55)(15,56)(16,53)(17,34)(18,35)(19,36)(20,33)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64)>;
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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,9,15,63)(6,10,16,64)(7,11,13,61)(8,12,14,62)(29,39,34,41)(30,40,35,42)(31,37,36,43)(32,38,33,44)(45,54,50,60)(46,55,51,57)(47,56,52,58)(48,53,49,59), (5,11)(6,12)(7,9)(8,10)(13,63)(14,64)(15,61)(16,62)(21,26)(22,27)(23,28)(24,25)(29,39)(30,40)(31,37)(32,38)(33,44)(34,41)(35,42)(36,43)(45,52)(46,49)(47,50)(48,51)(53,55)(54,56)(57,59)(58,60), (1,50,20,45)(2,46,17,51)(3,52,18,47)(4,48,19,49)(5,40,15,42)(6,43,16,37)(7,38,13,44)(8,41,14,39)(9,35,63,30)(10,31,64,36)(11,33,61,32)(12,29,62,34)(21,56,26,58)(22,59,27,53)(23,54,28,60)(24,57,25,55), (1,32)(2,29)(3,30)(4,31)(5,58)(6,59)(7,60)(8,57)(9,52)(10,49)(11,50)(12,51)(13,54)(14,55)(15,56)(16,53)(17,34)(18,35)(19,36)(20,33)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64) );
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)], [(1,28,20,23),(2,25,17,24),(3,26,18,21),(4,27,19,22),(5,9,15,63),(6,10,16,64),(7,11,13,61),(8,12,14,62),(29,39,34,41),(30,40,35,42),(31,37,36,43),(32,38,33,44),(45,54,50,60),(46,55,51,57),(47,56,52,58),(48,53,49,59)], [(5,11),(6,12),(7,9),(8,10),(13,63),(14,64),(15,61),(16,62),(21,26),(22,27),(23,28),(24,25),(29,39),(30,40),(31,37),(32,38),(33,44),(34,41),(35,42),(36,43),(45,52),(46,49),(47,50),(48,51),(53,55),(54,56),(57,59),(58,60)], [(1,50,20,45),(2,46,17,51),(3,52,18,47),(4,48,19,49),(5,40,15,42),(6,43,16,37),(7,38,13,44),(8,41,14,39),(9,35,63,30),(10,31,64,36),(11,33,61,32),(12,29,62,34),(21,56,26,58),(22,59,27,53),(23,54,28,60),(24,57,25,55)], [(1,32),(2,29),(3,30),(4,31),(5,58),(6,59),(7,60),(8,57),(9,52),(10,49),(11,50),(12,51),(13,54),(14,55),(15,56),(16,53),(17,34),(18,35),(19,36),(20,33),(21,40),(22,37),(23,38),(24,39),(25,41),(26,42),(27,43),(28,44),(45,61),(46,62),(47,63),(48,64)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 15 | 0 |
| 0 | 0 | 7 | 6 | 0 | 15 |
| 0 | 0 | 9 | 8 | 11 | 10 |
| 0 | 0 | 8 | 9 | 10 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 1 |
| 0 | 0 | 0 | 7 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 10 | 1 | 0 |
| 0 | 0 | 7 | 0 | 0 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 2 | 0 |
| 0 | 0 | 10 | 4 | 0 | 2 |
| 0 | 0 | 1 | 11 | 13 | 7 |
| 0 | 0 | 11 | 1 | 7 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 13 | 3 | 3 |
| 0 | 0 | 4 | 0 | 3 | 14 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,6,7,9,8,0,0,7,6,8,9,0,0,15,0,11,10,0,0,0,15,10,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,10,0,0,0,1,0,0,7,0,0,0,0,0,16,0,0,0,0,1,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,7,0,0,0,16,10,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,4,10,1,11,0,0,10,4,11,1,0,0,2,0,13,7,0,0,0,2,7,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,4,0,0,3,14,13,0,0,0,0,0,3,3,0,0,0,0,3,14] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | D4○D8 | Q8○D8 |
| kernel | C42.358C23 | C42.7C22 | C4×D8 | C4×Q16 | SD16⋊C4 | D4⋊D4 | D4.7D4 | D4.2D4 | Q8.D4 | D4⋊Q8 | C4.Q16 | C22.D8 | C23.48D4 | C23.33C23 | C22.36C24 | C22⋊C4 | C4⋊C4 | D4 | Q8 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{358}C_2^3 % in TeX
G:=Group("C4^2.358C2^3"); // GroupNames label
G:=SmallGroup(128,1856);
// by ID
G=gap.SmallGroup(128,1856);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,1018,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,d*e=e*d>;
// generators/relations