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G = C2×C4⋊SD16order 128 = 27

Direct product of C2 and C4⋊SD16

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

Aliases: C2×C4⋊SD16, C42.208D4, C42.317C23, Q81(C2×D4), (C2×Q8)⋊27D4, C44(C2×SD16), C4⋊C873C22, (C2×C4)⋊16SD16, (C4×Q8)⋊77C22, C4.63(C22×D4), C4.66(C4⋊D4), C4⋊C4.373C23, (C2×C8).305C23, (C2×C4).236C24, (C2×D4).46C23, C23.855(C2×D4), (C22×C4).791D4, D4⋊C474C22, C2.8(C22×SD16), (C22×SD16)⋊22C2, (C2×SD16)⋊73C22, (C2×Q8).357C23, C22.83(C2×SD16), C41D4.135C22, (C2×C42).805C22, (C22×C8).338C22, C22.496(C22×D4), C22.168(C4⋊D4), C22.114(C8⋊C22), (C22×C4).1526C23, (C22×D4).336C22, (C22×Q8).469C22, (C2×C4⋊C8)⋊36C2, (C2×C4×Q8)⋊34C2, C4.146(C2×C4○D4), C2.54(C2×C4⋊D4), C2.14(C2×C8⋊C22), (C2×D4⋊C4)⋊38C2, (C2×C4).1416(C2×D4), (C2×C41D4).21C2, (C2×C4).903(C4○D4), (C2×C4⋊C4).917C22, SmallGroup(128,1764)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C4⋊SD16
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C2×C4⋊SD16
C1C2C2×C4 — C2×C4⋊SD16
C1C23C2×C42 — C2×C4⋊SD16
C1C2C2C2×C4 — C2×C4⋊SD16

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

Subgroups: 652 in 292 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×12], D4 [×28], Q8 [×4], Q8 [×6], C23, C23 [×16], C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4 [×3], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×6], C2×Q8 [×3], C24 [×2], D4⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C41D4 [×4], C41D4 [×2], C22×C8 [×2], C2×SD16 [×8], C2×SD16 [×8], C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×D4⋊C4 [×2], C2×C4⋊C8, C4⋊SD16 [×8], C2×C4×Q8, C2×C41D4, C22×SD16 [×2], C2×C4⋊SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], SD16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×SD16 [×6], C8⋊C22 [×2], C22×D4 [×2], C2×C4○D4, C4⋊SD16 [×4], C2×C4⋊D4, C22×SD16, C2×C8⋊C22, C2×C4⋊SD16

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4R8A···8H
order12···222224···44···48···8
size11···188882···24···44···4

38 irreducible representations

dim1111111222224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4SD16C4○D4C8⋊C22
kernelC2×C4⋊SD16C2×D4⋊C4C2×C4⋊C8C4⋊SD16C2×C4×Q8C2×C41D4C22×SD16C42C22×C4C2×Q8C2×C4C2×C4C22
# reps1218112224842

Matrix representation of C2×C4⋊SD16 in GL5(𝔽17)

160000
01000
00100
000160
000016
,
10000
01200
0161600
000160
000016
,
10000
016000
01100
000107
00050
,
160000
01000
0161600
00010
000116

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,16,0,0,0,2,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,0,10,5,0,0,0,7,0],[16,0,0,0,0,0,1,16,0,0,0,0,16,0,0,0,0,0,1,1,0,0,0,0,16] >;

C2×C4⋊SD16 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes {\rm SD}_{16}
% in TeX

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

G:=SmallGroup(128,1764);
// by ID

G=gap.SmallGroup(128,1764);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,4037,1027,124]);
// 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=b^-1,d*c*d=c^3>;
// generators/relations

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