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G = (C2×C4)⋊9D8order 128 = 27

1st semidirect product of C2×C4 and D8 acting via D8/D4=C2

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

Aliases: (C2×C4)⋊9D8, (C2×D8)⋊6C4, C2.7(C4×D8), C4.83(C4×D4), C4⋊C4.312D4, D41(C22⋊C4), (C2×D4).205D4, C2.2(C4⋊D8), C4.3(C4⋊D4), (C22×D8).1C2, C22.39(C2×D8), C2.7(D8⋊C4), C2.5(C22⋊D8), C2.6(D4⋊D4), C23.770(C2×D4), (C22×C4).689D4, C22.153(C4×D4), C22.4Q1635C2, C22.95C22≀C2, C2.3(D4.2D4), C22.57(C4○D8), (C22×C8).37C22, C22.77(C8⋊C22), (C2×C42).279C22, C24.3C222C2, C22.116(C4⋊D4), (C22×C4).1366C23, C22.7C4216C2, C4.65(C22.D4), (C22×D4).461C22, C2.18(C23.23D4), (C2×C4×D4)⋊1C2, (C2×C8)⋊6(C2×C4), (C2×D4)⋊6(C2×C4), (C2×D4⋊C4)⋊5C2, (C2×C4).998(C2×D4), C4.13(C2×C22⋊C4), (C2×C4).563(C4○D4), (C2×C4⋊C4).767C22, (C2×C4).384(C22×C4), SmallGroup(128,611)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×C4)⋊9D8
C1C2C4C2×C4C22×C4C22×D4C2×C4×D4 — (C2×C4)⋊9D8
C1C2C2×C4 — (C2×C4)⋊9D8
C1C23C2×C42 — (C2×C4)⋊9D8
C1C2C2C22×C4 — (C2×C4)⋊9D8

Generators and relations for (C2×C4)⋊9D8
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=ab-1, dcd=c-1 >

Subgroups: 532 in 219 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×7], C2 [×6], C4 [×4], C4 [×6], C22 [×7], C22 [×26], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], D4 [×4], D4 [×12], C23, C23 [×18], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×5], D8 [×8], C22×C4 [×3], C22×C4 [×10], C2×D4 [×8], C2×D4 [×8], C24 [×2], D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C4×D4 [×4], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C23×C4, C22×D4 [×2], C22.7C42, C22.4Q16, C24.3C22, C2×D4⋊C4 [×2], C2×C4×D4, C22×D8, (C2×C4)⋊9D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], D8 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×D8, C4○D8, C8⋊C22 [×2], C23.23D4, C4×D8, D8⋊C4, C22⋊D8, D4⋊D4, C4⋊D8, D4.2D4, (C2×C4)⋊9D8

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N4O4P8A···8H
order12···22222224···44···4448···8
size11···14444882···24···4884···4

38 irreducible representations

dim111111112222224
type++++++++++++
imageC1C2C2C2C2C2C2C4D4D4D4D8C4○D4C4○D8C8⋊C22
kernel(C2×C4)⋊9D8C22.7C42C22.4Q16C24.3C22C2×D4⋊C4C2×C4×D4C22×D8C2×D8C4⋊C4C22×C4C2×D4C2×C4C2×C4C22C22
# reps111121182244442

Matrix representation of (C2×C4)⋊9D8 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
0160000
1600000
0001300
0013000
000010
000001
,
0160000
100000
0001600
001000
000033
0000143
,
100000
0160000
001000
0001600
0000160
000001

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

(C2×C4)⋊9D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_9D_8
% in TeX

G:=Group("(C2xC4):9D8");
// GroupNames label

G:=SmallGroup(128,611);
// by ID

G=gap.SmallGroup(128,611);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic

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

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