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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×SD16)⋊15C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4×Q8 — (C2×SD16)⋊15C4
 Lower central C1 — C2 — C2×C4 — (C2×SD16)⋊15C4
 Upper central C1 — C23 — C2×C42 — (C2×SD16)⋊15C4
 Jennings C1 — C2 — C2 — C22×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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, D4⋊C4, Q8⋊C4, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×C8, C2×SD16, C2×SD16, 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, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, 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 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(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 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)
(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 34)(10 37)(11 40)(12 35)(13 38)(14 33)(15 36)(16 39)(25 48)(26 43)(27 46)(28 41)(29 44)(30 47)(31 42)(32 45)(49 60)(50 63)(51 58)(52 61)(53 64)(54 59)(55 62)(56 57)
(1 30 17 54)(2 62 18 42)(3 28 19 52)(4 60 20 48)(5 26 21 50)(6 58 22 46)(7 32 23 56)(8 64 24 44)(9 27 34 51)(10 59 35 47)(11 25 36 49)(12 57 37 45)(13 31 38 55)(14 63 39 43)(15 29 40 53)(16 61 33 41)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4R 4S 4T 8A ··· 8H order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 4 ··· 4 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 SD16 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel (C2×SD16)⋊15C4 C22.7C42 C22.4Q16 C24.3C22 C2×D4⋊C4 C2×Q8⋊C4 C2×C4×Q8 C22×SD16 C2×SD16 C4⋊C4 C22×C4 C2×Q8 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 1 1 1 1 1 1 8 2 2 4 4 4 4 1 1

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

 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 12 0 0 0 5 5 0 0 0 0 0 1 0 0 0 0 8 16
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 8 16
,
 13 0 0 0 0 0 14 3 0 0 0 3 3 0 0 0 0 0 8 15 0 0 0 6 9

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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