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G = C42.235D4order 128 = 27

217th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.235D4, C42.351C23, Q8.Q818C2, Q16⋊C48C2, C4⋊C4.70C23, C4⋊C8.53C22, (C2×C8).44C23, Q8.8(C4○D4), Q8⋊D4.1C2, (C2×C4).315C24, C42.6C48C2, Q8.D419C2, C22⋊Q1615C2, (C4×D4).79C22, (C2×D4).93C23, C23.677(C2×D4), (C22×C4).455D4, SD16⋊C411C2, (C4×Q8).76C22, C2.D8.90C22, C8⋊C4.10C22, C4.Q8.19C22, C22⋊C8.28C22, (C2×Q8).379C23, (C2×Q16).59C22, D4⋊C4.34C22, C23.48D416C2, C4⋊D4.169C22, (C2×C42).842C22, Q8⋊C4.35C22, (C2×SD16).16C22, C23.46D4.1C2, C22.575(C22×D4), C22⋊Q8.174C22, C2.34(D8⋊C22), (C22×C4).1031C23, C4.4D4.131C22, C22.19(C8.C22), C42.C2.108C22, (C22×Q8).480C22, C2.116(C22.19C24), C23.36C23.21C2, (C2×C4×Q8)⋊40C2, C4.200(C2×C4○D4), (C2×C4).1223(C2×D4), C2.36(C2×C8.C22), (C2×C4⋊C4).941C22, SmallGroup(128,1849)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.235D4
C1C2C4C2×C4C42C4×Q8C2×C4×Q8 — C42.235D4
C1C2C2×C4 — C42.235D4
C1C22C2×C42 — C42.235D4
C1C2C2C2×C4 — C42.235D4

Generators and relations for C42.235D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b, dcd=c3 >

Subgroups: 332 in 193 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×14], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×20], D4 [×4], Q8 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8 [×4], C4×Q8 [×2], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×SD16 [×2], C2×Q16 [×2], C22×Q8, C42.6C4, SD16⋊C4 [×2], Q16⋊C4 [×2], Q8⋊D4, C22⋊Q16, Q8.D4 [×2], Q8.Q8 [×2], C23.46D4, C23.48D4, C2×C4×Q8, C23.36C23, C42.235D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8.C22, D8⋊C22, C42.235D4

Smallest permutation representation of C42.235D4
On 64 points
Generators in S64
(1 63 5 59)(2 49 6 53)(3 57 7 61)(4 51 8 55)(9 21 13 17)(10 47 14 43)(11 23 15 19)(12 41 16 45)(18 36 22 40)(20 38 24 34)(25 52 29 56)(26 60 30 64)(27 54 31 50)(28 62 32 58)(33 44 37 48)(35 46 39 42)
(1 37 29 11)(2 34 30 16)(3 39 31 13)(4 36 32 10)(5 33 25 15)(6 38 26 12)(7 35 27 9)(8 40 28 14)(17 57 42 50)(18 62 43 55)(19 59 44 52)(20 64 45 49)(21 61 46 54)(22 58 47 51)(23 63 48 56)(24 60 41 53)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 46)(18 41)(19 44)(20 47)(21 42)(22 45)(23 48)(24 43)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(49 62)(50 57)(51 60)(52 63)(53 58)(54 61)(55 64)(56 59)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I···4R4S4T4U8A8B8C8D
order12222224···44···44448888
size11112282···24···48888888

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C8.C22D8⋊C22
kernelC42.235D4C42.6C4SD16⋊C4Q16⋊C4Q8⋊D4C22⋊Q16Q8.D4Q8.Q8C23.46D4C23.48D4C2×C4×Q8C23.36C23C42C22×C4Q8C22C2
# reps11221122111122822

Matrix representation of C42.235D4 in GL6(𝔽17)

1620000
010000
00161500
001100
000101
001616160
,
400000
040000
0010015
00160161
00161601
0000016
,
100000
1160000
0039116
007060
0081710
00101617
,
100000
1160000
001000
00161600
000010
0010016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,2,1,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,16,16,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,15,1,1,16],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,3,7,8,10,0,0,9,0,1,16,0,0,11,6,7,1,0,0,6,0,10,7],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C42.235D4 in GAP, Magma, Sage, TeX

C_4^2._{235}D_4
% in TeX

G:=Group("C4^2.235D4");
// GroupNames label

G:=SmallGroup(128,1849);
// by ID

G=gap.SmallGroup(128,1849);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,521,304,4037,1027,124]);
// Polycyclic

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

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