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G = (C2×C8)⋊13D4  order 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, C8⋊2D4⋊4C2, C8⋊8D4⋊44C2, C8.120(C2×D4), C8.D4⋊4C2, (C2×D4).219D4, C4⋊C4.29C23, (C2×Q8).174D4, C23.78(C2×D4), (C2×C4).264C24, (C2×C8).256C23, (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, C23.36D4⋊42C2, C22.14(C4⋊D4), (C22×C4).986C23, (C22×C8).261C22, (C2×Q16).156C22, C22.524(C22×D4), D4⋊C4.130C22, C22.31C24⋊5C2, 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

Derived series
C1 — C2×C4 — (C2×C8)⋊13D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8)⋊13D4
Lower central
C1 — C2 — C2×C4 — (C2×C8)⋊13D4
Upper central
C1 — C22 — C2×C4○D4 — (C2×C8)⋊13D4
Jennings
C1 — C2 — C2 — C2×C4 — (C2×C8)⋊13D4

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, C8⋊8D4 [×4], C8⋊2D4 [×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

Generators and relations
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b3, dcd=c-1 >

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

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

Matrix representation ►G ⊆ 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] >;
 

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.Q8C8⋊8D4C8⋊2D4C8.D4C22.31C24C2×C8○D4C2×C4○D8C2×C8C2×D4C2×Q8C2×C4C2
# reps12142221143144

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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