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G = (C2×C4)⋊3D8order 128 = 27

2nd semidirect product of C2×C4 and D8 acting via D8/C4=C22

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

Aliases: (C2×C4)⋊3D8, (C2×C8).49D4, C4.76C22≀C2, (C2×D4).115D4, (C22×D8).5C2, C22.85(C2×D8), C2.14(C87D4), C2.14(C4⋊D8), C2.14(C82D4), C23.916(C2×D4), (C22×C4).150D4, C22.4Q1638C2, C4.72(C4.4D4), (C22×C8).77C22, C2.18(D4.2D4), C22.112(C4○D8), (C2×C42).368C22, (C22×D4).86C22, C22.238(C4⋊D4), C22.140(C8⋊C22), C24.3C2210C2, (C22×C4).1450C23, C4.76(C22.D4), C2.27(C23.10D4), (C2×C4⋊C8)⋊20C2, (C2×D4⋊C4)⋊15C2, (C2×C4).1045(C2×D4), (C2×C4).881(C4○D4), (C2×C4⋊C4).131C22, SmallGroup(128,786)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4)⋊3D8
C1C2C4C2×C4C22×C4C22×C8C2×D4⋊C4 — (C2×C4)⋊3D8
C1C2C22×C4 — (C2×C4)⋊3D8
C1C23C2×C42 — (C2×C4)⋊3D8
C1C2C2C22×C4 — (C2×C4)⋊3D8

Generators and relations for (C2×C4)⋊3D8
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=c-1 >

Subgroups: 480 in 180 conjugacy classes, 52 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×5], C22 [×3], C22 [×4], C22 [×20], C8 [×3], C2×C4 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×12], C23, C23 [×16], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], D8 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C24 [×2], D4⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C2×D8 [×6], C22×D4 [×2], C22.4Q16, C24.3C22 [×2], C2×D4⋊C4 [×2], C2×C4⋊C8, C22×D8, (C2×C4)⋊3D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, D8 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×D8, C4○D8, C8⋊C22 [×2], C23.10D4, C4⋊D8 [×2], D4.2D4 [×2], C87D4, C82D4, (C2×C4)⋊3D8

Smallest permutation representation of (C2×C4)⋊3D8
On 64 points
Generators in S64
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)
(1 51 17 27)(2 28 18 52)(3 53 19 29)(4 30 20 54)(5 55 21 31)(6 32 22 56)(7 49 23 25)(8 26 24 50)(9 37 45 64)(10 57 46 38)(11 39 47 58)(12 59 48 40)(13 33 41 60)(14 61 42 34)(15 35 43 62)(16 63 44 36)
(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 8)(2 7)(3 6)(4 5)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 24)(18 23)(19 22)(20 21)(33 38)(34 37)(35 36)(39 40)(41 53)(42 52)(43 51)(44 50)(45 49)(46 56)(47 55)(48 54)(57 60)(58 59)(61 64)(62 63)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111112222224
type+++++++++++
imageC1C2C2C2C2C2D4D4D4D8C4○D4C4○D8C8⋊C22
kernel(C2×C4)⋊3D8C22.4Q16C24.3C22C2×D4⋊C4C2×C4⋊C8C22×D8C2×C8C22×C4C2×D4C2×C4C2×C4C22C22
# reps1122112244642

Matrix representation of (C2×C4)⋊3D8 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
100000
010000
000100
001000
000040
0000713
,
1430000
14140000
0016000
0001600
000079
0000210
,
3140000
14140000
0016000
000100
000079
0000610

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,7,0,0,0,0,0,13],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,2,0,0,0,0,9,10],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,7,6,0,0,0,0,9,10] >;

(C2×C4)⋊3D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3D_8
% in TeX

G:=Group("(C2xC4):3D8");
// GroupNames label

G:=SmallGroup(128,786);
// by ID

G=gap.SmallGroup(128,786);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,1411,172,4037,2028,124]);
// 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,c*b*c^-1=b^-1,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations

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