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G = (C2xC4):9SD16order 128 = 27

1st semidirect product of C2xC4 and SD16 acting via SD16/C8=C2

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

Aliases: (C2xC4):9SD16, C4.17(C4xD4), C8:5(C22:C4), (C2xC8).276D4, (C2xSD16):16C4, C2.7(C8:8D4), C2.4(C8:5D4), C2.19(C4xSD16), C4.81(C4:D4), C4.1(C4.4D4), C22.181(C4xD4), C23.801(C2xD4), (C22xC4).558D4, C2.3(C8.12D4), C22.75(C4oD8), (C22xSD16).8C2, C22.72(C2xSD16), C22.37(C4:1D4), (C22xC8).492C22, (C22xD4).48C22, (C22xQ8).38C22, C22.143(C4:D4), (C22xC4).1408C23, C23.67C23:4C2, (C2xC42).1075C22, C24.3C22.10C2, C2.20(C24.3C22), (C2xC4xC8):33C2, (C2xC4.Q8):19C2, (C2xC8).185(C2xC4), (C2xQ8:C4):9C2, C4.36(C2xC22:C4), (C2xQ8).94(C2xC4), (C2xD4:C4).9C2, (C2xD4).109(C2xC4), (C2xC4).1353(C2xD4), (C2xC4:C4).89C22, (C2xC4).599(C4oD4), (C2xC4).422(C22xC4), SmallGroup(128,700)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — (C2xC4):9SD16
C1C2C22C2xC4C22xC4C22xC8C2xC4xC8 — (C2xC4):9SD16
C1C2C2xC4 — (C2xC4):9SD16
C1C23C2xC42 — (C2xC4):9SD16
C1C2C2C22xC4 — (C2xC4):9SD16

Generators and relations for (C2xC4):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, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C2.C42, C4xC8, D4:C4, Q8:C4, C4.Q8, C2xC42, C2xC22:C4, C2xC4:C4, C22xC8, C2xSD16, C2xSD16, C22xD4, C22xQ8, C24.3C22, C23.67C23, C2xC4xC8, C2xD4:C4, C2xQ8:C4, C2xC4.Q8, C22xSD16, (C2xC4):9SD16
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, SD16, C22xC4, C2xD4, C4oD4, C2xC22:C4, C4xD4, C4:D4, C4.4D4, C4:1D4, C2xSD16, C4oD8, C24.3C22, C4xSD16, C8:8D4, C8:5D4, C8.12D4, (C2xC4):9SD16

Smallest permutation representation of (C2xC4):9SD16
On 64 points
Generators in S64
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 24 33 55)(2 17 34 56)(3 18 35 49)(4 19 36 50)(5 20 37 51)(6 21 38 52)(7 22 39 53)(8 23 40 54)(9 57 32 42)(10 58 25 43)(11 59 26 44)(12 60 27 45)(13 61 28 46)(14 62 29 47)(15 63 30 48)(16 64 31 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 53)(42 56)(43 51)(44 54)(45 49)(46 52)(47 55)(48 50)

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

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

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

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++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4SD16C4oD4C4oD8
kernel(C2xC4):9SD16C24.3C22C23.67C23C2xC4xC8C2xD4:C4C2xQ8:C4C2xC4.Q8C22xSD16C2xSD16C2xC8C22xC4C2xC4C2xC4C22
# reps11111111862848

Matrix representation of (C2xC4):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] >;

(C2xC4):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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