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G = C23.39D8order 128 = 27

10th non-split extension by C23 of D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.39D8, C4○D82C4, D88(C2×C4), Q168(C2×C4), C8.93(C2×D4), (C2×C8).117D4, (C2×C4).137D8, C2.D1614C2, C8.4(C22⋊C4), C8.30(C22×C4), (C2×C4).47SD16, C4.11(C2×SD16), C22.54(C2×D8), C2.Q3214C2, C2.1(C16⋊C22), (C2×M5(2))⋊12C2, (C2×C8).495C23, (C2×C16).50C22, C2.1(Q32⋊C2), (C22×C4).330D4, C4.37(D4⋊C4), (C2×D8).101C22, C2.D8.144C22, C22.4(D4⋊C4), (C22×C8).229C22, (C2×Q16).100C22, (C2×C8).79(C2×C4), (C2×C4○D8).9C2, (C2×C2.D8)⋊35C2, (C2×C4).757(C2×D4), C4.51(C2×C22⋊C4), C2.29(C2×D4⋊C4), (C2×C4).148(C22⋊C4), SmallGroup(128,871)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.39D8
C1C2C4C2×C4C2×C8C22×C8C2×C4○D8 — C23.39D8
C1C2C4C8 — C23.39D8
C1C22C22×C4C22×C8 — C23.39D8
C1C2C2C2C2C4C4C2×C8 — C23.39D8

Generators and relations for C23.39D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd7 >

Subgroups: 276 in 116 conjugacy classes, 52 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, C16 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×4], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], C2.D8 [×2], C2.D8, C2×C16 [×2], M5(2) [×2], C2×C4⋊C4, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C2×C4○D4, C2.D16 [×2], C2.Q32 [×2], C2×C2.D8, C2×M5(2), C2×C4○D8, C23.39D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C16⋊C22, Q32⋊C2, C23.39D8

Smallest permutation representation of C23.39D8
On 64 points
Generators in S64
(1 43)(2 36)(3 45)(4 38)(5 47)(6 40)(7 33)(8 42)(9 35)(10 44)(11 37)(12 46)(13 39)(14 48)(15 41)(16 34)(17 54)(18 63)(19 56)(20 49)(21 58)(22 51)(23 60)(24 53)(25 62)(26 55)(27 64)(28 57)(29 50)(30 59)(31 52)(32 61)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(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 64 49 16)(2 15 50 63)(3 62 51 14)(4 13 52 61)(5 60 53 12)(6 11 54 59)(7 58 55 10)(8 9 56 57)(17 30 40 37)(18 36 41 29)(19 28 42 35)(20 34 43 27)(21 26 44 33)(22 48 45 25)(23 24 46 47)(31 32 38 39)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4J8A8B8C8D8E8F16A···16H
order1222222244444···488888816···16
size1111228822228···82222444···4

32 irreducible representations

dim11111112222244
type+++++++++++-
imageC1C2C2C2C2C2C4D4D4D8SD16D8C16⋊C22Q32⋊C2
kernelC23.39D8C2.D16C2.Q32C2×C2.D8C2×M5(2)C2×C4○D8C4○D8C2×C8C22×C4C2×C4C2×C4C23C2C2
# reps12211183124222

Matrix representation of C23.39D8 in GL6(𝔽17)

1600000
0160000
000010
000001
001000
000100
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
940000
1480000
0013762
0056164
001115410
001131211
,
940000
580000
0013762
001641211
0062137
001211164

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[9,14,0,0,0,0,4,8,0,0,0,0,0,0,13,5,11,1,0,0,7,6,15,13,0,0,6,16,4,12,0,0,2,4,10,11],[9,5,0,0,0,0,4,8,0,0,0,0,0,0,13,16,6,12,0,0,7,4,2,11,0,0,6,12,13,16,0,0,2,11,7,4] >;

C23.39D8 in GAP, Magma, Sage, TeX

C_2^3._{39}D_8
% in TeX

G:=Group("C2^3.39D8");
// GroupNames label

G:=SmallGroup(128,871);
// by ID

G=gap.SmallGroup(128,871);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^7>;
// generators/relations

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