direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×2- (1+4), C22.16C25, C42.537C23, C23.110C24, Q8○(C4×D4), D4○(C4×Q8), C4.42(C23×C4), C2.12(C24×C4), C4⋊C4.517C23, (C2×C4).162C24, D4.25(C22×C4), C22.6(C23×C4), Q8.26(C22×C4), (C4×D4).348C22, (C2×D4).500C23, (C2×Q8).483C23, (C4×Q8).318C22, C2.3(C2×2- (1+4)), C22⋊C4.129C23, C2.3(C2.C25), (C2×C42).914C22, (C2×2- (1+4)).8C2, (C22×C4).1296C23, (C22×Q8).483C22, C4○3(C23.33C23), C4○2(C23.32C23), C23.33C23⋊35C2, C23.32C23⋊21C2, C42⋊C2.336C22, C4⋊C4○(C4×Q8), (C4×D4)○(C4×Q8), (C2×D4)○(C4×Q8), (C2×C4×Q8)⋊43C2, C22⋊C4○(C4×Q8), (C4×C4○D4)⋊15C2, C4○D4⋊11(C2×C4), (C2×Q8)⋊30(C2×C4), (C2×C4).91(C22×C4), (C2×C4⋊C4).942C22, (C2×C4○D4).319C22, SmallGroup(128,2162)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 836 in 746 conjugacy classes, 686 normal (8 characteristic)
C1, C2 [×3], C2 [×10], C4 [×22], C4 [×15], C22, C22 [×10], C22 [×10], C2×C4, C2×C4 [×85], C2×C4 [×25], D4 [×40], Q8 [×40], C23 [×5], C42 [×40], C22⋊C4 [×20], C4⋊C4 [×40], C22×C4 [×35], C2×D4 [×10], C2×Q8 [×50], C4○D4 [×80], C2×C42 [×15], C2×C4⋊C4 [×15], C42⋊C2 [×30], C4×D4 [×40], C4×Q8 [×40], C22×Q8 [×5], C2×C4○D4 [×10], 2- (1+4) [×16], C2×C4×Q8 [×5], C4×C4○D4 [×10], C23.32C23 [×5], C23.33C23 [×10], C2×2- (1+4), C4×2- (1+4)
Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], 2- (1+4) [×2], C25, C24×C4, C2×2- (1+4), C2.C25, C4×2- (1+4)
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
►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 54 11 57)(2 55 12 58)(3 56 9 59)(4 53 10 60)(5 17 64 14)(6 18 61 15)(7 19 62 16)(8 20 63 13)(21 42 28 39)(22 43 25 40)(23 44 26 37)(24 41 27 38)(29 47 36 50)(30 48 33 51)(31 45 34 52)(32 46 35 49)
(1 14)(2 15)(3 16)(4 13)(5 57)(6 58)(7 59)(8 60)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)(25 35)(26 36)(27 33)(28 34)(37 47)(38 48)(39 45)(40 46)(41 51)(42 52)(43 49)(44 50)(53 63)(54 64)(55 61)(56 62)
(1 35 11 32)(2 36 12 29)(3 33 9 30)(4 34 10 31)(5 40 64 43)(6 37 61 44)(7 38 62 41)(8 39 63 42)(13 28 20 21)(14 25 17 22)(15 26 18 23)(16 27 19 24)(45 53 52 60)(46 54 49 57)(47 55 50 58)(48 56 51 59)
(1 59 11 56)(2 60 12 53)(3 57 9 54)(4 58 10 55)(5 19 64 16)(6 20 61 13)(7 17 62 14)(8 18 63 15)(21 37 28 44)(22 38 25 41)(23 39 26 42)(24 40 27 43)(29 45 36 52)(30 46 33 49)(31 47 34 50)(32 48 35 51)
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), (1,54,11,57)(2,55,12,58)(3,56,9,59)(4,53,10,60)(5,17,64,14)(6,18,61,15)(7,19,62,16)(8,20,63,13)(21,42,28,39)(22,43,25,40)(23,44,26,37)(24,41,27,38)(29,47,36,50)(30,48,33,51)(31,45,34,52)(32,46,35,49), (1,14)(2,15)(3,16)(4,13)(5,57)(6,58)(7,59)(8,60)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30)(25,35)(26,36)(27,33)(28,34)(37,47)(38,48)(39,45)(40,46)(41,51)(42,52)(43,49)(44,50)(53,63)(54,64)(55,61)(56,62), (1,35,11,32)(2,36,12,29)(3,33,9,30)(4,34,10,31)(5,40,64,43)(6,37,61,44)(7,38,62,41)(8,39,63,42)(13,28,20,21)(14,25,17,22)(15,26,18,23)(16,27,19,24)(45,53,52,60)(46,54,49,57)(47,55,50,58)(48,56,51,59), (1,59,11,56)(2,60,12,53)(3,57,9,54)(4,58,10,55)(5,19,64,16)(6,20,61,13)(7,17,62,14)(8,18,63,15)(21,37,28,44)(22,38,25,41)(23,39,26,42)(24,40,27,43)(29,45,36,52)(30,46,33,49)(31,47,34,50)(32,48,35,51)>;
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), (1,54,11,57)(2,55,12,58)(3,56,9,59)(4,53,10,60)(5,17,64,14)(6,18,61,15)(7,19,62,16)(8,20,63,13)(21,42,28,39)(22,43,25,40)(23,44,26,37)(24,41,27,38)(29,47,36,50)(30,48,33,51)(31,45,34,52)(32,46,35,49), (1,14)(2,15)(3,16)(4,13)(5,57)(6,58)(7,59)(8,60)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30)(25,35)(26,36)(27,33)(28,34)(37,47)(38,48)(39,45)(40,46)(41,51)(42,52)(43,49)(44,50)(53,63)(54,64)(55,61)(56,62), (1,35,11,32)(2,36,12,29)(3,33,9,30)(4,34,10,31)(5,40,64,43)(6,37,61,44)(7,38,62,41)(8,39,63,42)(13,28,20,21)(14,25,17,22)(15,26,18,23)(16,27,19,24)(45,53,52,60)(46,54,49,57)(47,55,50,58)(48,56,51,59), (1,59,11,56)(2,60,12,53)(3,57,9,54)(4,58,10,55)(5,19,64,16)(6,20,61,13)(7,17,62,14)(8,18,63,15)(21,37,28,44)(22,38,25,41)(23,39,26,42)(24,40,27,43)(29,45,36,52)(30,46,33,49)(31,47,34,50)(32,48,35,51) );
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)], [(1,54,11,57),(2,55,12,58),(3,56,9,59),(4,53,10,60),(5,17,64,14),(6,18,61,15),(7,19,62,16),(8,20,63,13),(21,42,28,39),(22,43,25,40),(23,44,26,37),(24,41,27,38),(29,47,36,50),(30,48,33,51),(31,45,34,52),(32,46,35,49)], [(1,14),(2,15),(3,16),(4,13),(5,57),(6,58),(7,59),(8,60),(9,19),(10,20),(11,17),(12,18),(21,31),(22,32),(23,29),(24,30),(25,35),(26,36),(27,33),(28,34),(37,47),(38,48),(39,45),(40,46),(41,51),(42,52),(43,49),(44,50),(53,63),(54,64),(55,61),(56,62)], [(1,35,11,32),(2,36,12,29),(3,33,9,30),(4,34,10,31),(5,40,64,43),(6,37,61,44),(7,38,62,41),(8,39,63,42),(13,28,20,21),(14,25,17,22),(15,26,18,23),(16,27,19,24),(45,53,52,60),(46,54,49,57),(47,55,50,58),(48,56,51,59)], [(1,59,11,56),(2,60,12,53),(3,57,9,54),(4,58,10,55),(5,19,64,16),(6,20,61,13),(7,17,62,14),(8,18,63,15),(21,37,28,44),(22,38,25,41),(23,39,26,42),(24,40,27,43),(29,45,36,52),(30,46,33,49),(31,47,34,50),(32,48,35,51)])
Matrix representation ►G ⊆ GL5(𝔽5)
| 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0,2,0,0,0,0,0,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,4,0,0,0,1,0,0,0,1,0,0,0],[4,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0,3,0,0,0,0,0,3,0,0] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4BB |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | 2- (1+4) | C2.C25 |
| kernel | C4×2- (1+4) | C2×C4×Q8 | C4×C4○D4 | C23.32C23 | C23.33C23 | C2×2- (1+4) | 2- (1+4) | C4 | C2 |
| # reps | 1 | 5 | 10 | 5 | 10 | 1 | 32 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4\times 2_-^{(1+4)} % in TeX
G:=Group("C4xES-(2,2)"); // GroupNames label
G:=SmallGroup(128,2162);
// by ID
G=gap.SmallGroup(128,2162);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,387,184,1123,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations