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G = (C2×C4)⋊9SD16order 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), C85(C22⋊C4), (C2×C8).276D4, (C2×SD16)⋊16C4, C2.7(C88D4), C2.4(C85D4), 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(C41D4), (C22×C8).492C22, (C22×D4).48C22, (C22×Q8).38C22, C22.143(C4⋊D4), (C22×C4).1408C23, C23.67C234C2, (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

C1C2×C4 — (C2×C4)⋊9SD16
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — (C2×C4)⋊9SD16
C1C2C2×C4 — (C2×C4)⋊9SD16
C1C23C2×C42 — (C2×C4)⋊9SD16
C1C2C2C22×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, C41D4, C2×SD16 [×2], C4○D8 [×2], C24.3C22, C4×SD16 [×2], C88D4 [×2], C85D4, 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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