Copied to
clipboard

G = C2×D43Q8order 128 = 27

Direct product of C2 and D43Q8

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

Aliases: C2×D43Q8, C22.61C25, C23.126C24, C24.618C23, C42.559C23, C22.1132+ 1+4, D46(C2×Q8), (C2×D4)⋊24Q8, C4⋊Q884C22, (C2×C4).60C24, (C4×Q8)⋊95C22, C2.11(Q8×C23), C4.49(C22×Q8), C4⋊C4.471C23, C22⋊Q889C22, C23.123(C2×Q8), (C4×D4).352C22, (C2×D4).502C23, C22.9(C22×Q8), C22⋊C4.85C23, (C2×Q8).279C23, C42.C249C22, (C23×C4).599C22, (C2×C42).930C22, C2.20(C2×2+ 1+4), (C22×C4).1197C23, (C22×D4).616C22, (C22×Q8).354C22, D43(C2×C4⋊C4), C4⋊C44(C2×D4), (C2×C4×Q8)⋊53C2, (C2×C4⋊Q8)⋊52C2, (C2×C4×D4).88C2, (C22×C4⋊C4)⋊46C2, C4.134(C2×C4○D4), (C2×C22⋊Q8)⋊73C2, (C2×C4).323(C2×Q8), (C2×C4⋊C4)⋊138C22, (C2×C42.C2)⋊44C2, C2.33(C22×C4○D4), (C2×C4).850(C4○D4), C22.159(C2×C4○D4), (C2×C22⋊C4).541C22, SmallGroup(128,2204)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×D43Q8
C1C2C22C23C22×C4C23×C4C22×C4⋊C4 — C2×D43Q8
C1C22 — C2×D43Q8
C1C23 — C2×D43Q8
C1C22 — C2×D43Q8

Generators and relations for C2×D43Q8
 G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b2c, ede-1=d-1 >

Subgroups: 828 in 600 conjugacy classes, 444 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C22×Q8, C22×C4⋊C4, C2×C4×D4, C2×C4×D4, C2×C4×Q8, C2×C22⋊Q8, C2×C42.C2, C2×C4⋊Q8, D43Q8, C2×D43Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, C25, D43Q8, Q8×C23, C22×C4○D4, C2×2+ 1+4, C2×D43Q8

Smallest permutation representation of C2×D43Q8
On 64 points
Generators in S64
(1 14)(2 15)(3 16)(4 13)(5 32)(6 29)(7 30)(8 31)(9 33)(10 34)(11 35)(12 36)(17 43)(18 44)(19 41)(20 42)(21 25)(22 26)(23 27)(24 28)(37 45)(38 46)(39 47)(40 48)(49 64)(50 61)(51 62)(52 63)(53 57)(54 58)(55 59)(56 60)
(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 16)(2 15)(3 14)(4 13)(5 32)(6 31)(7 30)(8 29)(9 33)(10 36)(11 35)(12 34)(17 43)(18 42)(19 41)(20 44)(21 27)(22 26)(23 25)(24 28)(37 45)(38 48)(39 47)(40 46)(49 64)(50 63)(51 62)(52 61)(53 59)(54 58)(55 57)(56 60)
(1 36 25 8)(2 33 26 5)(3 34 27 6)(4 35 28 7)(9 22 32 15)(10 23 29 16)(11 24 30 13)(12 21 31 14)(17 37 51 60)(18 38 52 57)(19 39 49 58)(20 40 50 59)(41 47 64 54)(42 48 61 55)(43 45 62 56)(44 46 63 53)
(1 47 25 54)(2 48 26 55)(3 45 27 56)(4 46 28 53)(5 61 33 42)(6 62 34 43)(7 63 35 44)(8 64 36 41)(9 20 32 50)(10 17 29 51)(11 18 30 52)(12 19 31 49)(13 38 24 57)(14 39 21 58)(15 40 22 59)(16 37 23 60)

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O4A···4P4Q···4AH
order12···22···24···44···4
size11···12···22···24···4

50 irreducible representations

dim11111111224
type++++++++-+
imageC1C2C2C2C2C2C2C2Q8C4○D42+ 1+4
kernelC2×D43Q8C22×C4⋊C4C2×C4×D4C2×C4×Q8C2×C22⋊Q8C2×C42.C2C2×C4⋊Q8D43Q8C2×D4C2×C4C22
# reps123162116882

Matrix representation of C2×D43Q8 in GL5(𝔽5)

40000
01000
00100
00010
00001
,
10000
04000
00400
00012
00044
,
10000
01000
00100
00040
00011
,
40000
00400
01000
00040
00004
,
40000
02000
00300
00024
00033

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,4,0,0,0,2,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,1,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,2,3,0,0,0,4,3] >;

C2×D43Q8 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_3Q_8
% in TeX

G:=Group("C2xD4:3Q8");
// GroupNames label

G:=SmallGroup(128,2204);
// by ID

G=gap.SmallGroup(128,2204);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,680,1430,570,136]);
// Polycyclic

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

׿
×
𝔽