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G = (C2×C4)⋊3SD16order 128 = 27

1st semidirect product of C2×C4 and SD16 acting via SD16/C4=C22

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

Aliases: (C2×C4)⋊3SD16, C4.9C22≀C2, (C2×D4).92D4, (C2×C8).154D4, (C2×Q8).83D4, C2.12(C85D4), C4.45(C4⋊D4), C23.901(C2×D4), (C22×C4).308D4, C2.19(Q8⋊D4), C2.13(C8.2D4), (C22×SD16).9C2, C22.91(C2×SD16), C2.12(D4.D4), C22.203C22≀C2, C2.18(C232D4), C22.74(C41D4), (C2×C42).349C22, (C22×C8).318C22, (C22×D4).65C22, (C22×Q8).54C22, C22.222(C4⋊D4), (C22×C4).1435C23, C22.7C4230C2, C22.118(C8.C22), C24.3C22.12C2, (C2×C4⋊Q8)⋊2C2, (C2×C4).745(C2×D4), (C2×Q8⋊C4)⋊33C2, (C2×C4).874(C4○D4), (C2×C4⋊C4).108C22, SmallGroup(128,745)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4)⋊3SD16
C1C2C22C2×C4C22×C4C22×Q8C2×C4⋊Q8 — (C2×C4)⋊3SD16
C1C2C22×C4 — (C2×C4)⋊3SD16
C1C23C2×C42 — (C2×C4)⋊3SD16
C1C2C2C22×C4 — (C2×C4)⋊3SD16

Generators and relations for (C2×C4)⋊3SD16
 G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=c3 >

Subgroups: 456 in 203 conjugacy classes, 58 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×6], Q8 [×12], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×4], C2×Q8 [×10], C24, Q8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4⋊Q8 [×4], C22×C8 [×2], C2×SD16 [×12], C22×D4, C22×Q8 [×2], C22.7C42, C24.3C22, C2×Q8⋊C4 [×2], C2×C4⋊Q8, C22×SD16 [×2], (C2×C4)⋊3SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, SD16 [×4], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×SD16 [×2], C8.C22 [×2], C232D4, Q8⋊D4 [×2], D4.D4 [×2], C85D4, C8.2D4, (C2×C4)⋊3SD16

Smallest permutation representation of (C2×C4)⋊3SD16
On 64 points
Generators in S64
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 60 27 40)(2 54 28 21)(3 62 29 34)(4 56 30 23)(5 64 31 36)(6 50 32 17)(7 58 25 38)(8 52 26 19)(9 37 43 57)(10 18 44 51)(11 39 45 59)(12 20 46 53)(13 33 47 61)(14 22 48 55)(15 35 41 63)(16 24 42 49)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 59)(18 62)(19 57)(20 60)(21 63)(22 58)(23 61)(24 64)(25 29)(26 32)(28 30)(33 56)(34 51)(35 54)(36 49)(37 52)(38 55)(39 50)(40 53)(41 47)(43 45)(44 48)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111112222224
type++++++++++-
imageC1C2C2C2C2C2D4D4D4D4SD16C4○D4C8.C22
kernel(C2×C4)⋊3SD16C22.7C42C24.3C22C2×Q8⋊C4C2×C4⋊Q8C22×SD16C2×C8C22×C4C2×D4C2×Q8C2×C4C2×C4C22
# reps1112124224822

Matrix representation of (C2×C4)⋊3SD16 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
400000
0130000
0001600
001000
0000016
0000160
,
010000
100000
0012500
00121200
000001
0000160
,
100000
010000
000100
001000
000010
0000016

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

(C2×C4)⋊3SD16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3{\rm SD}_{16}
% in TeX

G:=Group("(C2xC4):3SD16");
// GroupNames label

G:=SmallGroup(128,745);
// by ID

G=gap.SmallGroup(128,745);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,352,2019,1018,521,248,2804,1411,718,172]);
// 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^3>;
// generators/relations

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