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G = C2×C23.24D4order 128 = 27

Direct product of C2 and C23.24D4

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

Aliases: C2×C23.24D4, C24.139D4, (C23×C8)⋊6C2, C4.3(C23×C4), C4⋊C4.340C23, (C2×C8).467C23, (C2×C4).173C24, (C22×C8)⋊63C22, D4.17(C22×C4), C4.138(C22×D4), (C22×C4).819D4, C23.376(C2×D4), Q8.17(C22×C4), D4⋊C497C22, Q8⋊C499C22, (C2×D4).358C23, C22.81(C4○D8), C4(C23.24D4), (C2×Q8).331C23, C42⋊C273C22, (C23×C4).690C22, C22.123(C22×D4), C23.130(C22⋊C4), (C22×C4).1497C23, (C22×D4).551C22, (C22×Q8).455C22, C4(C2×D4⋊C4), C4(C2×Q8⋊C4), C2.1(C2×C4○D8), (C2×C4○D4)⋊18C4, C4○D412(C2×C4), (C2×C4)2(D4⋊C4), (C2×D4⋊C4)⋊58C2, (C2×C4)2(Q8⋊C4), (C2×Q8⋊C4)⋊59C2, (C2×D4).226(C2×C4), (C2×C4).1563(C2×D4), C4.121(C2×C22⋊C4), (C2×Q8).204(C2×C4), (C2×C42⋊C2)⋊40C2, (C22×C4)(D4⋊C4), (C2×C4⋊C4).899C22, (C2×C4).458(C22×C4), (C22×C4).415(C2×C4), (C22×C4)(Q8⋊C4), (C22×C4○D4).18C2, C22.20(C2×C22⋊C4), C2.35(C22×C22⋊C4), (C2×C4).284(C22⋊C4), (C2×C4○D4).272C22, (C2×C4)(C23.24D4), (C2×C4)(C2×D4⋊C4), (C2×C4)(C2×Q8⋊C4), (C22×C4)(C2×D4⋊C4), (C22×C4)(C2×Q8⋊C4), SmallGroup(128,1624)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C23.24D4
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C2×C23.24D4
C1C2C4 — C2×C23.24D4
C1C22×C4C23×C4 — C2×C23.24D4
C1C2C2C2×C4 — C2×C23.24D4

Generators and relations for C2×C23.24D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 668 in 396 conjugacy classes, 180 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×28], C8 [×4], C2×C4 [×2], C2×C4 [×26], C2×C4 [×30], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23, C23 [×6], C23 [×14], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×12], C22×C4 [×15], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×16], C4○D4 [×24], C24, C24, D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×6], C22×C8 [×4], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C23.24D4 [×8], C2×C42⋊C2, C23×C8, C22×C4○D4, C2×C23.24D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C4○D8 [×4], C23×C4, C22×D4 [×2], C23.24D4 [×4], C22×C22⋊C4, C2×C4○D8 [×2], C2×C23.24D4

Smallest permutation representation of C2×C23.24D4
On 64 points
Generators in S64
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(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 31 32 8)(2 7 25 30)(3 29 26 6)(4 5 27 28)(9 14 48 45)(10 44 41 13)(11 12 42 43)(15 16 46 47)(17 22 60 57)(18 64 61 21)(19 20 62 63)(23 24 58 59)(33 50 51 40)(34 39 52 49)(35 56 53 38)(36 37 54 55)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L4M···4X8A···8P
order12···2222222224···444444···48···8
size11···1222244441···122224···42···2

56 irreducible representations

dim11111111222
type+++++++++
imageC1C2C2C2C2C2C2C4D4D4C4○D8
kernelC2×C23.24D4C2×D4⋊C4C2×Q8⋊C4C23.24D4C2×C42⋊C2C23×C8C22×C4○D4C2×C4○D4C22×C4C24C22
# reps1228111167116

Matrix representation of C2×C23.24D4 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000004
0000130
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
0160000
1600000
0001600
001000
0000143
00001414
,
0160000
100000
0001600
0016000
0000143
000033

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

C2×C23.24D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{24}D_4
% in TeX

G:=Group("C2xC2^3.24D4");
// GroupNames label

G:=SmallGroup(128,1624);
// by ID

G=gap.SmallGroup(128,1624);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,1411,172]);
// Polycyclic

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

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