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G = (C2×C4)⋊9SD16  order 128 = 27

1st semidirect product of C2×C4 and SD16 acting via SD16/C8=C2

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

Aliases: (C2×C4)⋊9SD16, C4.17(C4×D4), C8⋊5(C22⋊C4), (C2×C8).276D4, (C2×SD16)⋊16C4, C2.7(C8⋊8D4), C2.4(C8⋊5D4), C2.19(C4×SD16), C4.81(C4⋊D4), C4.1(C4.4D4), C22.181(C4×D4), C23.801(C2×D4), (C22×C4).558D4, C2.3(C8.12D4), C22.75(C4○D8), (C22×SD16).8C2, C22.72(C2×SD16), C22.37(C4⋊1D4), (C22×C8).492C22, (C22×D4).48C22, (C22×Q8).38C22, C22.143(C4⋊D4), (C22×C4).1408C23, C23.67C23⋊4C2, (C2×C42).1075C22, C24.3C22.10C2, C2.20(C24.3C22), (C2×C4×C8)⋊33C2, (C2×C4.Q8)⋊19C2, (C2×C8).185(C2×C4), (C2×Q8⋊C4)⋊9C2, C4.36(C2×C22⋊C4), (C2×Q8).94(C2×C4), (C2×D4⋊C4).9C2, (C2×D4).109(C2×C4), (C2×C4).1353(C2×D4), (C2×C4⋊C4).89C22, (C2×C4).599(C4○D4), (C2×C4).422(C22×C4), SmallGroup(128,700)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — (C2×C4)⋊9SD16
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — (C2×C4)⋊9SD16
Lower central
C1 — C2 — C2×C4 — (C2×C4)⋊9SD16
Upper central
C1 — C23 — C2×C42 — (C2×C4)⋊9SD16
Jennings
C1 — C2 — C2 — C22×C4 — (C2×C4)⋊9SD16

Generators and relations for (C2×C4)⋊9SD16
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=ab-1, dcd=c3 >

Subgroups: 388 in 174 conjugacy classes, 68 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×14], D4 [×6], Q8 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42 [×2], C4×C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C2×SD16 [×4], C2×SD16 [×4], C22×D4, C22×Q8, C24.3C22, C23.67C23, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C22×SD16, (C2×C4)⋊9SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], SD16 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C4⋊1D4, C2×SD16 [×2], C4○D8 [×2], C24.3C22, C4×SD16 [×2], C8⋊8D4 [×2], C8⋊5D4, C8.12D4, (C2×C4)⋊9SD16

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4R8A···8P
order12···2224···44···48···8
size11···1882···28···82···2

44 irreducible representations

dim11111111122222
type++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4SD16C4○D4C4○D8
kernel(C2×C4)⋊9SD16C24.3C22C23.67C23C2×C4×C8C2×D4⋊C4C2×Q8⋊C4C2×C4.Q8C22×SD16C2×SD16C2×C8C22×C4C2×C4C2×C4C22
# reps11111111862848

Matrix representation of (C2×C4)⋊9SD16 ►in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
400000
040000
001000
000100
0000415
00001613
,
0100000
12100000
00101000
0012000
0000160
0000016
,
1150000
0160000
0016000
001100
000010
0000416

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,16,0,0,0,0,15,13],[0,12,0,0,0,0,10,10,0,0,0,0,0,0,10,12,0,0,0,0,10,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,16] >;
 

(C2×C4)⋊9SD16 in GAP, Magma, Sage, TeX 

(C_2\times C_4)\rtimes_9{\rm SD}_{16}
 
% in TeX
 
G:=Group("(C2xC4):9SD16");
 
// GroupNames label
 
G:=SmallGroup(128,700);
 
// by ID
 
G=gap.SmallGroup(128,700);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,2804,172]);
 
// 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,b*c=c*b,d*b*d=a*b^-1,d*c*d=c^3>;
 
// generators/relations
 

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