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G = (C2×C8)⋊20D4order 128 = 27

16th semidirect product of C2×C8 and D4 acting via D4/C2=C22

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

Aliases: (C2×C8)⋊20D4, (C2×Q8)⋊11D4, (C2×C4)⋊4SD16, (C2×D4).93D4, C4.10C22≀C2, C2.15(C83D4), C2.13(C85D4), C4.46(C4⋊D4), C23.902(C2×D4), (C22×C4).309D4, C2.12(C4⋊SD16), (C22×SD16)⋊13C2, C22.92(C2×SD16), C2.19(C22⋊SD16), C22.204C22≀C2, C2.19(C232D4), C22.75(C41D4), (C22×C8).319C22, (C2×C42).350C22, (C22×D4).66C22, (C22×Q8).55C22, C22.223(C4⋊D4), C22.131(C8⋊C22), (C22×C4).1436C23, C22.7C4231C2, C23.67C237C2, (C2×C4).746(C2×D4), (C2×C41D4).9C2, (C2×D4⋊C4)⋊34C2, (C2×C4).875(C4○D4), (C2×C4⋊C4).109C22, SmallGroup(128,746)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8)⋊20D4
C1C2C22C2×C4C22×C4C22×D4C2×C41D4 — (C2×C8)⋊20D4
C1C2C22×C4 — (C2×C8)⋊20D4
C1C23C2×C42 — (C2×C8)⋊20D4
C1C2C2C22×C4 — (C2×C8)⋊20D4

Generators and relations for (C2×C8)⋊20D4
 G = < a,b,c,d | a2=b8=c4=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=b3, dcd=c-1 >

Subgroups: 616 in 237 conjugacy classes, 58 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], D4 [×28], Q8 [×6], C23, C23 [×16], C42 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×2], C2×Q8 [×5], C24 [×2], C2.C42 [×2], D4⋊C4 [×4], C2×C42, C2×C4⋊C4, C41D4 [×4], C22×C8 [×2], C2×SD16 [×12], C22×D4 [×2], C22×D4 [×2], C22×Q8, C22.7C42, C23.67C23, C2×D4⋊C4 [×2], C2×C41D4, C22×SD16 [×2], (C2×C8)⋊20D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, SD16 [×4], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×SD16 [×2], C8⋊C22 [×2], C232D4, C22⋊SD16 [×2], C4⋊SD16 [×2], C85D4, C83D4, (C2×C8)⋊20D4

Smallest permutation representation of (C2×C8)⋊20D4
On 64 points
Generators in S64
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(41 55)(42 56)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(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 33 21 45)(2 28 22 52)(3 35 23 47)(4 30 24 54)(5 37 17 41)(6 32 18 56)(7 39 19 43)(8 26 20 50)(9 51 59 27)(10 46 60 34)(11 53 61 29)(12 48 62 36)(13 55 63 31)(14 42 64 38)(15 49 57 25)(16 44 58 40)
(1 45)(2 48)(3 43)(4 46)(5 41)(6 44)(7 47)(8 42)(9 27)(10 30)(11 25)(12 28)(13 31)(14 26)(15 29)(16 32)(17 37)(18 40)(19 35)(20 38)(21 33)(22 36)(23 39)(24 34)(49 61)(50 64)(51 59)(52 62)(53 57)(54 60)(55 63)(56 58)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L8A···8H
order12···222224444444444448···8
size11···188882222444488884···4

32 irreducible representations

dim1111112222224
type+++++++++++
imageC1C2C2C2C2C2D4D4D4D4SD16C4○D4C8⋊C22
kernel(C2×C8)⋊20D4C22.7C42C23.67C23C2×D4⋊C4C2×C41D4C22×SD16C2×C8C22×C4C2×D4C2×Q8C2×C4C2×C4C22
# reps1112124242822

Matrix representation of (C2×C8)⋊20D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
5140000
3120000
000700
005700
00001212
0000512
,
010000
100000
001000
000100
000001
0000160
,
010000
100000
001000
0011600
000001
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,3,0,0,0,0,14,12,0,0,0,0,0,0,0,5,0,0,0,0,7,7,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

(C2×C8)⋊20D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{20}D_4
% in TeX

G:=Group("(C2xC8):20D4");
// GroupNames label

G:=SmallGroup(128,746);
// by ID

G=gap.SmallGroup(128,746);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

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

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