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G = C2×C23.38D4order 128 = 27

Direct product of C2 and C23.38D4

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

Aliases: C2×C23.38D4, C24.169D4, C4.5(C23×C4), (C22×Q8)⋊21C4, C4⋊C4.341C23, (C2×C4).175C24, (C2×C8).390C23, C4.140(C22×D4), (C22×C4).779D4, C23.638(C2×D4), Q8.18(C22×C4), (Q8×C23).11C2, Q8⋊C486C22, (C2×Q8).332C23, (C22×C4).899C23, (C22×C8).423C22, (C23×C4).513C22, C22.125(C22×D4), C23.208(C22⋊C4), C22.93(C8.C22), (C22×M4(2)).26C2, (C22×Q8).456C22, C42⋊C2.281C22, (C2×M4(2)).333C22, (C2×Q8)⋊38(C2×C4), C4.74(C2×C22⋊C4), C2.1(C2×C8.C22), (C2×Q8⋊C4)⋊49C2, (C2×C4).1405(C2×D4), (C2×C4⋊C4).900C22, (C22×C4).323(C2×C4), (C2×C4).460(C22×C4), C22.81(C2×C22⋊C4), C2.37(C22×C22⋊C4), (C2×C4).157(C22⋊C4), (C2×C42⋊C2).50C2, SmallGroup(128,1626)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C23.38D4
C1C2C22C2×C4C22×C4C23×C4Q8×C23 — C2×C23.38D4
C1C2C4 — C2×C23.38D4
C1C23C23×C4 — C2×C23.38D4
C1C2C2C2×C4 — C2×C23.38D4

Generators and relations for C2×C23.38D4
 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, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 588 in 376 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×26], C2×C4 [×36], Q8 [×8], Q8 [×28], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×16], C2×Q8 [×28], C2×Q8 [×42], C24, Q8⋊C4 [×16], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4, C22×Q8 [×14], C22×Q8 [×7], C2×Q8⋊C4 [×4], C23.38D4 [×8], C2×C42⋊C2, C22×M4(2), Q8×C23, C2×C23.38D4
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], C8.C22 [×4], C23×C4, C22×D4 [×2], C23.38D4 [×4], C22×C22⋊C4, C2×C8.C22 [×2], C2×C23.38D4

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X8A···8H
order12···222224···44···48···8
size11···122222···24···44···4

44 irreducible representations

dim1111111224
type++++++++-
imageC1C2C2C2C2C2C4D4D4C8.C22
kernelC2×C23.38D4C2×Q8⋊C4C23.38D4C2×C42⋊C2C22×M4(2)Q8×C23C22×Q8C22×C4C24C22
# reps14811116714

Matrix representation of C2×C23.38D4 in GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
00000010
000041301
,
10000000
01000000
001600000
000160000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
016000000
10000000
000130000
00400000
000011649
00000040
000013000
0000130016
,
01000000
10000000
000130000
001300000
00000010
0000134115
000016000
000000013

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,4,0,0,0,0,0,16,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,13,13,0,0,0,0,16,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,9,0,0,16],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,13,16,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,13] >;

C2×C23.38D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{38}D_4
% in TeX

G:=Group("C2xC2^3.38D4");
// GroupNames label

G:=SmallGroup(128,1626);
// by ID

G=gap.SmallGroup(128,1626);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,1430,2804,1411,172]);
// 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,e*b*e^-1=f*b*f^-1=b*d=d*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*e^3>;
// generators/relations

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