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G = (C2×C8)⋊14D4order 128 = 27

10th semidirect product of C2×C8 and D4 acting via D4/C2=C22

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

Aliases: (C2×C8)⋊14D4, C8⋊D44C2, C87D435C2, C8.111(C2×D4), (C2×D4).220D4, C2.11(D4○D8), C4⋊C4.30C23, (C2×Q8).175D4, C2.11(Q8○D8), C23.79(C2×D4), C8.18D435C2, (C2×C4).265C24, (C2×C8).257C23, (C2×D4).67C23, C4.159(C22×D4), (C2×Q8).55C23, C4.175(C4⋊D4), (C2×D8).120C22, C4⋊D4.21C22, C2.D8.163C22, C22⋊Q8.21C22, C23.36D443C2, C22.15(C4⋊D4), (C22×C4).987C23, (C22×C8).262C22, (C2×Q16).118C22, C22.525(C22×D4), D4⋊C4.131C22, C22.31C246C2, Q8⋊C4.124C22, (C2×SD16).113C22, (C2×M4(2)).267C22, (C2×C8○D4)⋊4C2, (C2×C4○D8)⋊18C2, (C2×C2.D8)⋊41C2, C4.32(C2×C4○D4), (C2×C4).133(C2×D4), C2.83(C2×C4⋊D4), (C2×C4).286(C4○D4), (C2×C4⋊C4).594C22, (C2×C4○D4).128C22, SmallGroup(128,1793)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×C8)⋊14D4
C1C2C22C2×C4C22×C4C2×C4○D4C2×C8○D4 — (C2×C8)⋊14D4
C1C2C2×C4 — (C2×C8)⋊14D4
C1C22C2×C4○D4 — (C2×C8)⋊14D4
C1C2C2C2×C4 — (C2×C8)⋊14D4

Generators and relations for (C2×C8)⋊14D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 476 in 242 conjugacy classes, 100 normal (24 characteristic)
C1, C2 [×3], 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], C2.D8 [×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×C2.D8, C87D4 [×2], C8.18D4 [×2], C8⋊D4 [×4], C22.31C24 [×2], C2×C8○D4, C2×C4○D8, (C2×C8)⋊14D4
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○D8, Q8○D8, (C2×C8)⋊14D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111111222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4D4○D8Q8○D8
kernel(C2×C8)⋊14D4C23.36D4C2×C2.D8C87D4C8.18D4C8⋊D4C22.31C24C2×C8○D4C2×C4○D8C2×C8C2×D4C2×Q8C2×C4C2C2
# reps121224211431422

Matrix representation of (C2×C8)⋊14D4 in GL6(𝔽17)

100000
010000
000001
0000160
0001600
001000
,
100000
010000
0031400
003300
0000314
000033
,
2160000
5150000
00512116
0012121616
00116125
00161655
,
100000
4160000
0031400
00141400
0000314
00001414

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[2,5,0,0,0,0,16,15,0,0,0,0,0,0,5,12,1,16,0,0,12,12,16,16,0,0,1,16,12,5,0,0,16,16,5,5],[1,4,0,0,0,0,0,16,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,14,14] >;

(C2×C8)⋊14D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{14}D_4
% in TeX

G:=Group("(C2xC8):14D4");
// GroupNames label

G:=SmallGroup(128,1793);
// by ID

G=gap.SmallGroup(128,1793);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,521,248,2804,172]);
// 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^-1,d*c*d=c^-1>;
// generators/relations

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