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G = (C2×C8)⋊13D4order 128 = 27

9th semidirect product of C2×C8 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: (C2×C8)⋊13D4, C82D44C2, C88D444C2, C8.120(C2×D4), C8.D44C2, (C2×D4).219D4, C4⋊C4.29C23, (C2×Q8).174D4, C23.78(C2×D4), (C2×C8).256C23, (C2×C4).264C24, (C2×D4).66C23, C4.158(C22×D4), (C2×Q8).54C23, C4.174(C4⋊D4), C2.17(D4○SD16), (C2×D8).161C22, C4⋊D4.20C22, C4.Q8.144C22, C22⋊Q8.20C22, C23.36D442C2, C22.14(C4⋊D4), (C22×C4).986C23, (C22×C8).261C22, (C2×Q16).156C22, C22.524(C22×D4), D4⋊C4.130C22, C22.31C245C2, Q8⋊C4.123C22, (C2×SD16).135C22, (C2×M4(2)).266C22, (C2×C8○D4)⋊3C2, (C2×C4○D8)⋊17C2, (C2×C4.Q8)⋊11C2, C4.31(C2×C4○D4), (C2×C4).132(C2×D4), C2.82(C2×C4⋊D4), (C2×C4).285(C4○D4), (C2×C4⋊C4).593C22, (C2×C4○D4).127C22, SmallGroup(128,1792)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×C8)⋊13D4
C1C2C22C2×C4C22×C4C2×C4○D4C2×C8○D4 — (C2×C8)⋊13D4
C1C2C2×C4 — (C2×C8)⋊13D4
C1C22C2×C4○D4 — (C2×C8)⋊13D4
C1C2C2C2×C4 — (C2×C8)⋊13D4

Generators and relations for (C2×C8)⋊13D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b3, dcd=c-1 >

Subgroups: 476 in 242 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×18], Q8 [×6], C23, C23 [×2], C23 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C4○D4 [×12], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C4⋊C4 [×2], C4⋊D4 [×4], C4⋊D4 [×6], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C2×C4○D4, C2×C4○D4 [×2], C23.36D4 [×2], C2×C4.Q8, C88D4 [×4], C82D4 [×2], C8.D4 [×2], C22.31C24 [×2], C2×C8○D4, C2×C4○D8, (C2×C8)⋊13D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○SD16 [×2], (C2×C8)⋊13D4

Smallest permutation representation of (C2×C8)⋊13D4
On 64 points
Generators in S64
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(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 46 55 63)(2 41 56 58)(3 44 49 61)(4 47 50 64)(5 42 51 59)(6 45 52 62)(7 48 53 57)(8 43 54 60)(9 24 32 40)(10 19 25 35)(11 22 26 38)(12 17 27 33)(13 20 28 36)(14 23 29 39)(15 18 30 34)(16 21 31 37)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 33)(18 36)(19 39)(20 34)(21 37)(22 40)(23 35)(24 38)(25 29)(26 32)(28 30)(41 64)(42 59)(43 62)(44 57)(45 60)(46 63)(47 58)(48 61)(49 53)(50 56)(52 54)

G:=sub<Sym(64)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (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), (1,46,55,63)(2,41,56,58)(3,44,49,61)(4,47,50,64)(5,42,51,59)(6,45,52,62)(7,48,53,57)(8,43,54,60)(9,24,32,40)(10,19,25,35)(11,22,26,38)(12,17,27,33)(13,20,28,36)(14,23,29,39)(15,18,30,34)(16,21,31,37), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,33)(18,36)(19,39)(20,34)(21,37)(22,40)(23,35)(24,38)(25,29)(26,32)(28,30)(41,64)(42,59)(43,62)(44,57)(45,60)(46,63)(47,58)(48,61)(49,53)(50,56)(52,54)>;

G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (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), (1,46,55,63)(2,41,56,58)(3,44,49,61)(4,47,50,64)(5,42,51,59)(6,45,52,62)(7,48,53,57)(8,43,54,60)(9,24,32,40)(10,19,25,35)(11,22,26,38)(12,17,27,33)(13,20,28,36)(14,23,29,39)(15,18,30,34)(16,21,31,37), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,33)(18,36)(19,39)(20,34)(21,37)(22,40)(23,35)(24,38)(25,29)(26,32)(28,30)(41,64)(42,59)(43,62)(44,57)(45,60)(46,63)(47,58)(48,61)(49,53)(50,56)(52,54) );

G=PermutationGroup([(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)], [(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)], [(1,46,55,63),(2,41,56,58),(3,44,49,61),(4,47,50,64),(5,42,51,59),(6,45,52,62),(7,48,53,57),(8,43,54,60),(9,24,32,40),(10,19,25,35),(11,22,26,38),(12,17,27,33),(13,20,28,36),(14,23,29,39),(15,18,30,34),(16,21,31,37)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,33),(18,36),(19,39),(20,34),(21,37),(22,40),(23,35),(24,38),(25,29),(26,32),(28,30),(41,64),(42,59),(43,62),(44,57),(45,60),(46,63),(47,58),(48,61),(49,53),(50,56),(52,54)])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4L8A8B8C8D8E···8J
order12222222224444444···488888···8
size11112244882222448···822224···4

32 irreducible representations

dim11111111122224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4D4○SD16
kernel(C2×C8)⋊13D4C23.36D4C2×C4.Q8C88D4C82D4C8.D4C22.31C24C2×C8○D4C2×C4○D8C2×C8C2×D4C2×Q8C2×C4C2
# reps12142221143144

Matrix representation of (C2×C8)⋊13D4 in GL6(𝔽17)

1600000
0160000
001002
001011
00161016
0000016
,
100000
010000
000700
005700
0005125
001251212
,
0160000
100000
008302
00010161
0010609
001071016
,
100000
0160000
001000
0011600
000010
00160016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,1,16,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,12,0,0,7,7,5,5,0,0,0,0,12,12,0,0,0,0,5,12],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,8,0,10,10,0,0,3,10,6,7,0,0,0,16,0,10,0,0,2,1,9,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

(C2×C8)⋊13D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{13}D_4
% in TeX

G:=Group("(C2xC8):13D4");
// GroupNames label

G:=SmallGroup(128,1792);
// by ID

G=gap.SmallGroup(128,1792);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,521,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^4,c*b*c^-1=d*b*d=b^3,d*c*d=c^-1>;
// generators/relations

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