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G = (C2×C4)⋊2D8order 128 = 27

1st semidirect product of C2×C4 and D8 acting via D8/C4=C22

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

Aliases: (C2×C4)⋊2D8, (C2×C8)⋊6D4, (C2×D4)⋊10D4, C4.7C22≀C2, (C22×D8)⋊2C2, C22.81(C2×D8), C2.13(C83D4), C2.12(C4⋊D8), C4.43(C4⋊D4), C2.10(C84D4), (C22×C4).307D4, C23.899(C2×D4), C2.19(C22⋊D8), C22.201C22≀C2, C22.72(C41D4), C2.16(C232D4), (C22×C8).107C22, C24.3C227C2, (C2×C42).347C22, (C22×D4).63C22, C22.220(C4⋊D4), C22.129(C8⋊C22), C22.7C4218C2, (C22×C4).1433C23, (C2×C41D4)⋊2C2, (C2×C4).744(C2×D4), (C2×D4⋊C4)⋊19C2, (C2×C4).872(C4○D4), (C2×C4⋊C4).106C22, SmallGroup(128,743)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4)⋊2D8
C1C2C22C2×C4C22×C4C22×D4C2×C41D4 — (C2×C4)⋊2D8
C1C2C22×C4 — (C2×C4)⋊2D8
C1C23C2×C42 — (C2×C4)⋊2D8
C1C2C2C22×C4 — (C2×C4)⋊2D8

Generators and relations for (C2×C4)⋊2D8
 G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=c-1 >

Subgroups: 712 in 255 conjugacy classes, 58 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×5], C22 [×3], C22 [×4], C22 [×30], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×7], D4 [×34], C23, C23 [×24], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], D8 [×16], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×6], C2×D4 [×31], C24 [×3], D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C41D4 [×4], C22×C8 [×2], C2×D8 [×12], C22×D4, C22×D4 [×2], C22×D4 [×2], C22.7C42, C24.3C22, C2×D4⋊C4 [×2], C2×C41D4, C22×D8 [×2], (C2×C4)⋊2D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, D8 [×4], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×D8 [×2], C8⋊C22 [×2], C232D4, C22⋊D8 [×2], C4⋊D8 [×2], C84D4, C83D4, (C2×C4)⋊2D8

Smallest permutation representation of (C2×C4)⋊2D8
On 64 points
Generators in S64
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(1 48 51 57)(2 34 52 20)(3 42 53 59)(4 36 54 22)(5 44 55 61)(6 38 56 24)(7 46 49 63)(8 40 50 18)(9 62 28 45)(10 17 29 39)(11 64 30 47)(12 19 31 33)(13 58 32 41)(14 21 25 35)(15 60 26 43)(16 23 27 37)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(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)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (1,48,51,57)(2,34,52,20)(3,42,53,59)(4,36,54,22)(5,44,55,61)(6,38,56,24)(7,46,49,63)(8,40,50,18)(9,62,28,45)(10,17,29,39)(11,64,30,47)(12,19,31,33)(13,58,32,41)(14,21,25,35)(15,60,26,43)(16,23,27,37), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(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( (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (1,48,51,57)(2,34,52,20)(3,42,53,59)(4,36,54,22)(5,44,55,61)(6,38,56,24)(7,46,49,63)(8,40,50,18)(9,62,28,45)(10,17,29,39)(11,64,30,47)(12,19,31,33)(13,58,32,41)(14,21,25,35)(15,60,26,43)(16,23,27,37), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(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([(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,53),(26,54),(27,55),(28,56),(29,49),(30,50),(31,51),(32,52),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)], [(1,48,51,57),(2,34,52,20),(3,42,53,59),(4,36,54,22),(5,44,55,61),(6,38,56,24),(7,46,49,63),(8,40,50,18),(9,62,28,45),(10,17,29,39),(11,64,30,47),(12,19,31,33),(13,58,32,41),(14,21,25,35),(15,60,26,43),(16,23,27,37)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,34),(18,33),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(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 2A···2G2H···2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12···22···244444444448···8
size11···18···822224444884···4

32 irreducible representations

dim111111222224
type+++++++++++
imageC1C2C2C2C2C2D4D4D4D8C4○D4C8⋊C22
kernel(C2×C4)⋊2D8C22.7C42C24.3C22C2×D4⋊C4C2×C41D4C22×D8C2×C8C22×C4C2×D4C2×C4C2×C4C22
# reps111212426822

Matrix representation of (C2×C4)⋊2D8 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
0011600
0021600
000001
0000160
,
330000
1430000
0016000
0001600
000010
0000016
,
14140000
1430000
0016000
0015100
000010
000001

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

(C2×C4)⋊2D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_2D_8
% in TeX

G:=Group("(C2xC4):2D8");
// GroupNames label

G:=SmallGroup(128,743);
// by ID

G=gap.SmallGroup(128,743);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

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

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