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G = (C2×SD16)⋊15C4order 128 = 27

11st semidirect product of C2×SD16 and C4 acting via C4/C2=C2

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

Aliases: C4.84(C4×D4), C4⋊C4.313D4, (C2×C4)⋊15SD16, (C2×SD16)⋊15C4, C4.4(C4⋊D4), Q81(C22⋊C4), (C2×Q8).163D4, C2.11(C4×SD16), C2.7(D4⋊D4), C2.5(Q8⋊D4), C2.2(C4⋊SD16), C23.771(C2×D4), (C22×C4).690D4, C22.154(C4×D4), C22.4Q1641C2, C22.96C22≀C2, C22.58(C4○D8), (C22×C8).38C22, C2.3(Q8.D4), (C22×SD16).7C2, C22.59(C2×SD16), C22.78(C8⋊C22), (C2×C42).280C22, (C22×D4).17C22, C2.11(SD16⋊C4), C22.117(C4⋊D4), (C22×C4).1367C23, C22.7C4228C2, C4.66(C22.D4), C22.67(C8.C22), C24.3C22.5C2, (C22×Q8).388C22, C2.19(C23.23D4), (C2×C4×Q8)⋊1C2, (C2×C8).109(C2×C4), (C2×Q8⋊C4)⋊6C2, (C2×D4).73(C2×C4), (C2×C4).999(C2×D4), C4.14(C2×C22⋊C4), (C2×D4⋊C4).4C2, (C2×Q8).146(C2×C4), (C2×C4).564(C4○D4), (C2×C4⋊C4).768C22, (C2×C4).385(C22×C4), SmallGroup(128,612)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×SD16)⋊15C4
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — (C2×SD16)⋊15C4
C1C2C2×C4 — (C2×SD16)⋊15C4
C1C23C2×C42 — (C2×SD16)⋊15C4
C1C2C2C22×C4 — (C2×SD16)⋊15C4

Generators and relations for (C2×SD16)⋊15C4
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=ab-1, dcd-1=ab6c >

Subgroups: 404 in 187 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×10], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×18], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×8], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], SD16 [×8], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×6], C2×Q8 [×3], C24, D4⋊C4 [×2], Q8⋊C4 [×2], C2×C42, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4×Q8 [×4], C22×C8 [×2], C2×SD16 [×4], C2×SD16 [×4], C22×D4, C22×Q8, C22.7C42, C22.4Q16, C24.3C22, C2×D4⋊C4, C2×Q8⋊C4, C2×C4×Q8, C22×SD16, (C2×SD16)⋊15C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], SD16 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.23D4, C4×SD16, SD16⋊C4, Q8⋊D4, D4⋊D4, C4⋊SD16, Q8.D4, (C2×SD16)⋊15C4

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4R4S4T8A···8H
order12···2224···44···4448···8
size11···1882···24···4884···4

38 irreducible representations

dim11111111122222244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4D4SD16C4○D4C4○D8C8⋊C22C8.C22
kernel(C2×SD16)⋊15C4C22.7C42C22.4Q16C24.3C22C2×D4⋊C4C2×Q8⋊C4C2×C4×Q8C22×SD16C2×SD16C4⋊C4C22×C4C2×Q8C2×C4C2×C4C22C22C22
# reps11111111822444411

Matrix representation of (C2×SD16)⋊15C4 in GL5(𝔽17)

10000
016000
001600
000160
000016
,
10000
051200
05500
00010
000816
,
10000
00100
01000
00010
000816
,
130000
014300
03300
000815
00069

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,1,8,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,16],[13,0,0,0,0,0,14,3,0,0,0,3,3,0,0,0,0,0,8,6,0,0,0,15,9] >;

(C2×SD16)⋊15C4 in GAP, Magma, Sage, TeX

(C_2\times {\rm SD}_{16})\rtimes_{15}C_4
% in TeX

G:=Group("(C2xSD16):15C4");
// GroupNames label

G:=SmallGroup(128,612);
// by ID

G=gap.SmallGroup(128,612);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,352,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=a*b^-1,d*c*d^-1=a*b^6*c>;
// generators/relations

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