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G = C2×C42.12C4order 128 = 27

Direct product of C2 and C42.12C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C42.12C4, C42.675C23, C422(C4⋊C8), C4⋊C893C22, (C4×C8)⋊67C22, (C22×C4)⋊10C8, C2.3(C23×C8), (C23×C4).41C4, C4.30(C22×C8), (C2×C42).55C4, C23.40(C2×C8), C422(C22⋊C8), C42.300(C2×C4), (C2×C8).472C23, C24.128(C2×C4), (C2×C4).636C24, (C2×C4).94M4(2), C4.70(C2×M4(2)), C4(C42.12C4), C22.39(C23×C4), (C22×C42).32C2, C22.14(C22×C8), C2.5(C22×M4(2)), C4.80(C42⋊C2), C22⋊C8.243C22, (C23×C4).657C22, (C22×C8).506C22, C23.223(C22×C4), C22.62(C2×M4(2)), C42(C42.12C4), (C22×C4).1270C23, (C2×C42).1104C22, C22.69(C42⋊C2), C42(C2×C4⋊C8), (C2×C4×C8)⋊15C2, C4⋊C8(C2×C42), (C2×C4⋊C8)⋊54C2, (C2×C4)3(C4⋊C8), (C2×C4)⋊12(C2×C8), C422(C2×C4⋊C8), C42(C2×C22⋊C8), C22⋊C8(C2×C42), (C2×C4)3(C22⋊C8), C4.287(C2×C4○D4), C422(C2×C22⋊C8), (C2×C22⋊C8).50C2, C2.5(C2×C42⋊C2), (C2×C4).952(C4○D4), (C22×C4).458(C2×C4), (C2×C4).626(C22×C4), (C2×C4)(C42.12C4), (C2×C4)2(C2×C4⋊C8), (C2×C42)(C2×C4⋊C8), SmallGroup(128,1649)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C42.12C4
C1C2C4C2×C4C22×C4C2×C42C22×C42 — C2×C42.12C4
C1C2 — C2×C42.12C4
C1C2×C42 — C2×C42.12C4
C1C2C2C2×C4 — C2×C42.12C4

Generators and relations for C2×C42.12C4
 G = < a,b,c,d | a2=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, cd=dc >

Subgroups: 332 in 264 conjugacy classes, 196 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C4 [×4], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×42], C2×C4 [×20], C23, C23 [×6], C23 [×4], C42 [×16], C2×C8 [×8], C2×C8 [×8], C22×C4 [×6], C22×C4 [×20], C22×C4 [×8], C24, C4×C8 [×8], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×4], C2×C42 [×8], C22×C8 [×4], C23×C4 [×3], C2×C4×C8 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C42.12C4 [×8], C22×C42, C2×C42.12C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], M4(2) [×4], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C22×C8 [×14], C2×M4(2) [×6], C23×C4, C2×C4○D4 [×2], C42.12C4 [×4], C2×C42⋊C2, C23×C8, C22×M4(2), C2×C42.12C4

Smallest permutation representation of C2×C42.12C4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 3 5 7)(2 18 6 22)(4 20 8 24)(9 11 13 15)(10 44 14 48)(12 46 16 42)(17 19 21 23)(25 27 29 31)(26 50 30 54)(28 52 32 56)(33 35 37 39)(34 58 38 62)(36 60 40 64)(41 43 45 47)(49 51 53 55)(57 59 61 63)
(1 57 19 39)(2 58 20 40)(3 59 21 33)(4 60 22 34)(5 61 23 35)(6 62 24 36)(7 63 17 37)(8 64 18 38)(9 25 45 51)(10 26 46 52)(11 27 47 53)(12 28 48 54)(13 29 41 55)(14 30 42 56)(15 31 43 49)(16 32 44 50)
(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)

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

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

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

80 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AJ8A···8AF
order12···222224···44···48···8
size11···122221···12···22···2

80 irreducible representations

dim11111111122
type++++++
imageC1C2C2C2C2C2C4C4C8M4(2)C4○D4
kernelC2×C42.12C4C2×C4×C8C2×C22⋊C8C2×C4⋊C8C42.12C4C22×C42C2×C42C23×C4C22×C4C2×C4C2×C4
# reps1222811243288

Matrix representation of C2×C42.12C4 in GL4(𝔽17) generated by

16000
01600
0010
0001
,
4000
01600
00160
0001
,
13000
01600
00130
00013
,
8000
01600
00016
0010
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[13,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[8,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0] >;

C2×C42.12C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{12}C_4
% in TeX

G:=Group("C2xC4^2.12C4");
// GroupNames label

G:=SmallGroup(128,1649);
// by ID

G=gap.SmallGroup(128,1649);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,124]);
// Polycyclic

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

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