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G = (C2×C4).27D8order 128 = 27

20th non-split extension by C2×C4 of D8 acting via D8/C4=C22

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

Aliases: (C2×C4).27D8, (C2×C8).59D4, (C2×D4).15Q8, C429C47C2, C22.92(C2×D8), C2.11(C84D4), C23.940(C2×D4), (C22×C4).324D4, C4.40(C22⋊Q8), C2.10(D4⋊Q8), C2.16(C8.2D4), C22.79(C41D4), (C22×C8).123C22, (C2×C42).391C22, (C22×D4).96C22, C2.6(C23.4Q8), C22.7C4219C2, (C22×C4).1474C23, C4.34(C22.D4), C2.10(C22.D8), C22.117(C22⋊Q8), C22.147(C8.C22), C24.3C22.22C2, C22.127(C22.D4), (C2×C2.D8)⋊14C2, (C2×C4).749(C2×D4), (C2×C4).291(C2×Q8), (C2×D4⋊C4).23C2, (C2×C4).780(C4○D4), (C2×C4⋊C4).163C22, SmallGroup(128,825)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4).27D8
C1C2C22C2×C4C22×C4C22×D4C24.3C22 — (C2×C4).27D8
C1C2C22×C4 — (C2×C4).27D8
C1C23C2×C42 — (C2×C4).27D8
C1C2C2C22×C4 — (C2×C4).27D8

Generators and relations for (C2×C4).27D8
 G = < a,b,c,d | a2=b4=c8=1, d2=ab2, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 352 in 145 conjugacy classes, 54 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, D4⋊C4 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×D4, C22.7C42, C429C4, C24.3C22, C2×D4⋊C4 [×2], C2×C2.D8 [×2], (C2×C4).27D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D8 [×4], C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C2×D8 [×2], C8.C22 [×2], C23.4Q8, D4⋊Q8 [×2], C22.D8 [×2], C84D4, C8.2D4, (C2×C4).27D8

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim111111222224
type++++++++-+-
imageC1C2C2C2C2C2D4D4Q8D8C4○D4C8.C22
kernel(C2×C4).27D8C22.7C42C429C4C24.3C22C2×D4⋊C4C2×C2.D8C2×C8C22×C4C2×D4C2×C4C2×C4C22
# reps111122422862

Matrix representation of (C2×C4).27D8 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
010000
1600000
0011500
0011600
000010
000001
,
16100000
1010000
0061100
003000
000033
0000143
,
7160000
16100000
0001100
0014000
000033
0000314

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],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,10,0,0,0,0,10,1,0,0,0,0,0,0,6,3,0,0,0,0,11,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[7,16,0,0,0,0,16,10,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14] >;

(C2×C4).27D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{27}D_8
% in TeX

G:=Group("(C2xC4).27D8");
// GroupNames label

G:=SmallGroup(128,825);
// by ID

G=gap.SmallGroup(128,825);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,624,422,387,1018,521,248,1411,718,172]);
// Polycyclic

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

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