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G = C4⋊2- 1+4order 128 = 27

The semidirect product of C4 and 2- 1+4 acting via 2- 1+4/C2×Q8=C2

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

Aliases: C412- 1+4, C23.42C24, C22.86C25, C42.574C23, (C2×Q8)⋊29D4, (D4×Q8)⋊17C2, Q82(C4⋊D4), Q8.60(C2×D4), Q85D416C2, Q86D417C2, C2.32(D4×C23), C4⋊C4.488C23, (C2×C4).603C24, C4.121(C22×D4), C4⋊Q8.339C22, C22.4(C22×D4), (C4×D4).230C22, (C2×D4).302C23, C22⋊C4.21C23, (C2×2- 1+4)⋊8C2, (C4×Q8).326C22, (C2×Q8).446C23, C41D4.185C22, C4⋊D4.114C22, (C22×C4).359C23, (C2×C42).942C22, C22⋊Q8.113C22, C2.22(C2×2- 1+4), C2.23(C2.C25), C22.26C2435C2, C4.4D4.174C22, (C22×Q8).358C22, C22.31C2416C2, (C2×C4×Q8)⋊59C2, (C2×Q8)(C4⋊D4), (C2×C4).187(C2×D4), (C2×C4⋊C4).963C22, (C2×C4○D4).227C22, SmallGroup(128,2229)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊2- 1+4
C1C2C22C23C22×C4C2×C42C2×C4×Q8 — C4⋊2- 1+4
C1C22 — C4⋊2- 1+4
C1C22 — C4⋊2- 1+4
C1C22 — C4⋊2- 1+4

Generators and relations for C4⋊2- 1+4
 G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 1108 in 746 conjugacy classes, 430 normal (12 characteristic)
C1, C2 [×3], C2 [×10], C4 [×14], C4 [×15], C22, C22 [×2], C22 [×26], C2×C4 [×2], C2×C4 [×32], C2×C4 [×53], D4 [×60], Q8 [×16], Q8 [×24], C23, C23 [×8], C42 [×12], C22⋊C4 [×32], C4⋊C4 [×28], C22×C4, C22×C4 [×30], C2×D4 [×36], C2×Q8 [×24], C2×Q8 [×32], C4○D4 [×80], C2×C42 [×3], C2×C4⋊C4 [×3], C4×D4 [×24], C4×Q8 [×8], C4⋊D4 [×40], C22⋊Q8 [×24], C4.4D4 [×12], C41D4 [×6], C4⋊Q8 [×6], C22×Q8, C22×Q8 [×8], C2×C4○D4 [×20], 2- 1+4 [×16], C2×C4×Q8, C22.26C24 [×6], C22.31C24 [×6], Q85D4 [×8], D4×Q8 [×4], Q86D4 [×4], C2×2- 1+4 [×2], C4⋊2- 1+4
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2- 1+4 [×2], C25, D4×C23, C2×2- 1+4, C2.C25, C4⋊2- 1+4

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A···4P4Q···4AD
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim11111111244
type+++++++++-
imageC1C2C2C2C2C2C2C2D42- 1+4C2.C25
kernelC4⋊2- 1+4C2×C4×Q8C22.26C24C22.31C24Q85D4D4×Q8Q86D4C2×2- 1+4C2×Q8C4C2
# reps11668442822

Matrix representation of C4⋊2- 1+4 in GL6(𝔽5)

010000
400000
001000
000100
000010
000001
,
400000
010000
002000
000300
000030
000232
,
400000
040000
000100
001000
001413
000004
,
400000
040000
001413
000010
000400
001404
,
400000
040000
003000
000300
000020
003202

G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,2,0,0,0,0,3,3,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,1,0,4,0,0,0,0,0,1,0,0,0,0,0,3,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,4,0,4,4,0,0,1,1,0,0,0,0,3,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,3,0,0,0,3,0,2,0,0,0,0,2,0,0,0,0,0,0,2] >;

C4⋊2- 1+4 in GAP, Magma, Sage, TeX

C_4\rtimes 2_-^{1+4}
% in TeX

G:=Group("C4:ES-(2,2)");
// GroupNames label

G:=SmallGroup(128,2229);
// by ID

G=gap.SmallGroup(128,2229);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,352,570,136,1684]);
// Polycyclic

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

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